Shupeng Zhao

Bernhard Rauer

Lorenzo Valzania

Jonathan Dong

Ruifeng Liu

Fuli Li

Sylvain Gigan

Hilton B De Aguiar
email: h.aguiar@lkb.ens.fr.

Single-pixel transmission matrix recovery via 2-photon fluorescence

Imaging at depth in opaque materials has long been a challenge. Recently, wavefront shaping has enabled significant advance for deep imaging. Nevertheless, most non-invasive wavefront shaping methods require cameras, lack the sensitivity for deep imaging under weak optical signals, or can only focus on a single "guidestar". Here, we retrieve the transmission matrix (TM) noninvasively using two-photon fluorescence exploiting a general single-pixel detection framework, allowing to achieve single-target focus on multiple guidestars spread beyond the memory effect range. In addition, if we assume memory effect correlations exist in the transmission matrix, we are able to significantly reduce the number of measurements needed.

Introduction

Fluorescence-based microscopy is the workhorse imaging modality in biology and medicine.

However, imaging opaque biological specimens at-depth is a current challenge in optics (1,2) as light passing through biological tissue experiences an environment of heterogeneous refractive index, leading to aberrations and scattering. To overcome these effects, feedback-based wavefront shaping techniques have emerged as a new paradigm to inverse the effect of scattering and focus light to a diffraction-limited spot (3). This is achieved by iteratively modulating the input wavefront with a spatial light modulator (SLM) aided by a feedback signal from the targeted point. However, this correction is only valid for one focus position and the process has to be repeated for another position. Retrieving the transmission matrix (TM), which connects input and output planes of the scattering medium, brings more freedom to focus light at any position of the target plane (4). Despite significant advances in imaging by characterizing the transmission matrix of a system, there are many applications scenarios where this cannot be neatly done by simply imaging the output target plane. Non-invasive settings required in biological imaging, typically using epi-geometry, are more challenging as they require an undisturbed feedback from within the medium for optimization-or TM-based schemes. In such cases, wavefront optimization requires the assistance of beacons or so-called "guidestars" in the target plane (5). Photoacoustic feedback (6,7), and acousto-optic tagging (8)(9)(10)(11)(12)(13) are compelling methods because of acoustic waves's resilience to scattering (13), which enables the reconstruction of a TM but requires complex super-resolution methods to enhance the resolution that is limited by the sound wavelength (14)(15)(16)(17). Another powerful method relies on the measurement of a time-gated matrix in reflection (18,19), but is based on retro-reflected ballistic excitation photons, hence limited in depth because of their exponential attenuation.

Fluorescence-based feedback signals can be easily separated from the excitation wave-length, however reach focusing at the optical diffraction limit has been a major challenge (5).

Because the fluorescence signal is strongly scattered by the complex medium, the main difficulty in wavefront shaping experiments is how to assess whether the excitation energy was delivered to only one guidestar. Optical non-linear fluorescence (20), or other computational non-linear methods (21)(22)(23) have been proposed to solve this problem. Once achieved focusing on a single guidestar, memory effect (ME) of the scattering medium (24,25) can be used for raster-scanning imaging (20). Nevertheless, the imaging range is limited by the ME range. A potential solution to overcome this limitation is to recover the TM using camera-based methods, which have been used to demix the emission signal from the different guidestars (26,27).

Summarizing current status (2), previous fluorescence based strategies can either focus at one single position using single-pixel detectors (12,20,(28)(29)(30)(31)(32) (or cameras (21,22)), or multiple positions by reconstructing the TM using cameras (26,27,33).Thus, a method based on singlepixel detection that can reconstruct the TM is urgently needed, as camera-based methods lack sensitivity at-depth (due to weaker signal generation).

Here, we propose a general model of fluorescence microscopy to perform non-invasive focusing and imaging at multiple target positions through the retrieval of the coherent TM, using only a single-pixel detector. We start by establishing a general framework and show that the forward model is only solvable with non-linear fluorescence signals. In particular, we use a multiplexed phase retrieval scheme using gradient-descent-based algorithm to extract the TM.

We demonstrate focusing on multiple fluorescing targets beyond the memory effect. To speed up the measurement process, we consider the memory effect which significantly decreases the amount of measurements needed. We solve for T using as input the measurements y and the wavefront modulation embedded in the user-input matrix A. The reconstruction of T allows to focus on each guidestar in the object. (c) In case 2, T has correlations, i.e. the infinite ME range is considered, and is given by the pixel-wise product of a rank-1 matrix T and a "tilt matrix" F T , where F is a matrix corresponding to 2D Fourier transform. An iterative method is used where, in each iteration, we solve for T at first, then the wavefronts estimation are used for raster-scanning the sample to update o until convergence.

Principle

We are interested in the situation described in Fig 1a : an extended incoherent object hidden behind a scattering medium is illuminated by a series of random speckle patterns, created by modulating an incident laser via spatial light modulator. The total fluorescence is integrated by a single-pixel detector. That is, we do not resolve the spatial fluctuations of the fluorescence pattern. We can model this in a general matrix-form phase-retrieval framework, where the detected signal y ∈ R Nm×1 is described by

y = |AT| 2p o, (1)
where A ∈ C Nm×N slm is a known measurement matrix, with each of the N m row representing a realization of modulated wavefront defined by N slm pixels, T ∈ C N slm ×N obj is a field transmission matrix from the SLM to the fluorescent object plane, o ∈ R N obj ×1 , and p the order of the p-photon excited fluorescence signal. In what follows, we consider two extreme cases: thick scattering medium with negligible ME range (the columns in T are fully decorrelated, case 1 in Fig. 1b) and thin scattering medium with infinite ME range (the columns in T are fully correlated, case 2 in Fig. 1c).

The object being sparse, only the positions where the fluorescent response of the object is non-zero contribute to the signal. Therefore we can consider only some columns of T where the fluorescent response is non-zero. To do that, we merge the vector o into the matrix T and Eq. ( 1) becomes:

y = N obj i=1 |At i | 2p , (2)
where N obj < N obj represents the number of columns left after merged with o. t i represent the columns of the transmission matrix T weighted by the corresponding elements in o 1 2p . Note that p = 1 corresponds to linear fluorescence, and in this case the solution of Eq.( 2) is not unique [START_REF] Dong | ICASSP 2019-2019 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)[END_REF], because there are infinite matrices T that satisfy the same measurement y (see the Supplementary Information for detail). Conversely, for p = 2 the ambiguity is lifted and Eq.( 2) has unique solution provided that a certain number of measurements is fulfilled (see the Supplementary Information for a mathematical proof). Therefore, as long as we detect nonlinear fluorescence signals and collect enough random wavefront measurements, the transmission matrix T can be uniquely determined. Note that A is, in our case, obtained by displaying a series of random patterns on a SLM, but could be achieved by any other deterministic modulation scheme (galvanometric mirrors, digital micromirror devices etc).

In order to reconstruct T with 2-photon fluorescence signal, we use a gradient-descent-based algorithm to solve Eq.( 2) when p = 2 (see methods for further details on the Algorithm 1). Since we are solving a multiplexed phase retrieval problem, the number of wavefronts measured N m should scale with the number of controlled modes N slm (the dimension of unknown signal) and the number of excitable speckles grains at the object plane where the fluorescence response is nonzero: N obj (the number of unknown signals). It means the required measurement number depends on the complexity of the object.

While this method allows to find a solution, including the presence of memory effect into our forward model allows us to reduce the measurement number. As showing in Fig. 1c, under the infinite memory effect condition, there is only phase ramp difference between columns of T. These phase ramp can be modeled by a "tilt matrix" F T , where F is a matrix corresponding to 2D Fourier transform (Multiplying F with a vector is equivalent to first reshaping the vector into a matrix, then doing 2D Fourier transform, and finally reshaping it back into a vector.). We can then rewrite Eq.( 1) to

y = |A(F T T )| 4 o, (3)
where T is a matrix that every column is the same vector t that corresponds to a wavefront ( stands for element-wise product). In order to retrieve both T and o in Eq.

Results

To demonstrate the feasibility of the methods, we have performed initial experiments on sparse fluorescence samples. We modulate the wavefront sequence using a SLM to define measurement matrix A (see method for details on the setup). The SLM is located conjugate to the back focal plane of the illumination objective. The fluorescence samples are drop-cast on a glass coverslip whose bottom surface, facing the illumination objective, was sand-blasted to introduce a single strongly scattering layer [START_REF] Rauer | [END_REF]. After the scattering medium and the sample, the 2PF signal y is collected by a photomultiplier tube (PMT). An additional charge-coupled device (CCD) camera is used only to inspect the quality of the focusing performance. A typical realization of the detected 2PF signal is shown in Fig. 2a where the fluctuations are due to the random wavefront realizations: on short-term averages these fluctuations are constant not showing any bleaching, and are furthermore well contrasted. We then solve Eq.( 2) when p = 2, from which we retrieve T to find the wavefronts arg(t * i ) to focus on individual multiple targets (Fig. 2b), or focii summed (Fig. 2c) as inspected by the CCD camera. Bright field images of the sources match well with the results (Fig. 2d). Residual correlations in T allows for imaging without raster-scanning the focus (Fig. 1d) and beyond the memory effect range (the red dashed circle in Fig. 2c represents the memory effect range). For this imaging example, we calculate the correlation of each two columns of T. Computing successively these pairwise correlations between close targets allows retrieving the full object, well beyond a single ME range (see Supplementary Information for further details). Importantly, we did not use any assumption on the between sources. To speed up measurement and reconstruction time, we then studied a system that has ME in T, and explicitly take it into account in the algorithm. Fig. 3 shows the experimental result when 8 random patterns are used for raster-scanning one by one to generate the measurement matrix

A. The initialization of o is (Fig. 3a1) obtained by raster-scanning the object without correcting for the scattering layer. In each iteration, we solve Eq.(3) first, from which we retrieve T , itself containing the wavefront t. Then, arg(t * ) is displayed on the SLM to correct the scattering layer for raster-scanning to obtain an update of o (Fig. 3a2). This process repeats until a desired stop criteria is achieved (in our case, a strong signal). Typically 3 iterations are needed to converge to a single focus as shown by the inspection camera at each iteration (Fig. 3b1-3). To ensure fast convergence, the dimension of o is set equal to N slm . Comparison of a high-resolution imaging taken after the optimization procedure (Fig. 3c2) shows excellent agreement with the inspection camera (Fig. 3c3). Inspection at each iteration shows that the speckle converges to a single foci despite the size of the object. Fig. 4 shows the experimental results for extended objects. Fig. 4a-b are fluorescence-dyed fibres into orthogonal directions spanning the whole speckle envelope. Be-fore optimization, clear replica of the object are seen because of the sparse 2PF speckle excitation. After optimization, convergence to a single focus is ensured. Even in extreme cases where the sample extend quasi homogeneously the speckle envelope (Fig. 4c), there is clearly a convergence to a single focus as shown by comparison with brightfield images (Fig. 4 c3, control camera). In these examples, because we reconstruct an extended complex o, to speed up convergence, we use 16 random patterns raster-scanning to generate A which weakly scales with the object complexity. The three results showing here take 3, 3 and 5 iterations to converge, respectively.

Discussion

We presented a method to non-invasively extract the transmission matrix of a buried fluorescent object, using a single-pixel detector. To validate the principle, we conducted an experiment by placing a single-pixel detector on one side of the object. However, in practice, it can also be placed on the other side to enable epi-detection mode. In both scenarios, the forward model of the system remains the same. Based on a phase-retrieval formalism, we exploit two extreme cases on the level of correlations embedded in T. If no correlations exists (as it would be the case for a very optically-thick medium), we show that retrieval of T is unique for nonlinear optical processes, such as two-photon fluorescence signals. The downside of this approach is that it requires a number of measurements that scales with the object complexity, which leads to lengthy acquisitions. We can largely circumvent this limitation if T displays correlations, in the form of memory effect, by introducing a tilt matrix F T . Rigorous solution is achieved by updating o at each new estimation of T. In this case, the number of wavefront measurements does not scale with object sparsity, so measurements can be significantly faster.

Two kind of methods exist for two-photon imaging through scattering medium. The first is known as wavefront shaping based method (12,36), which need fast SLMs in order to ulti-mately outpace the speckle decorrelation time imposed by living tissue. The second is focus scanning holographic aberration probing (F-SHARP) (29) and can be viewed as an interferometric method. Both, and works that follows (30,31), are optimization based methods and can only image within the memory effect range. We propose a method for imaging beyond the memory effect range, although it needs more measurements than optimization based methods, the data processing is not needed between the measurements which leaves a great possibility to improve the measurement speed. In addition, due to the random illumination strategy, the proposed method can prevent bleach compared to optimization based methods. A particular advantage here is that, when the imaging object is smaller than the ME range, the proposed method can realize faster focusing speed compared to wavefront shaping methods as in principle galvanometric and digital micromirror devices can be used, and have better stability compared to interferometric methods.

Methods

Experimental setup and sample preparation

The experimental setup is sketched in Fig. 8. The source used is an OPO (Coherent Mira OPO-X) pumped by a pulsed Ti-sapphire laser (Coherent Chameleon Ultra II) with 140 fs pulse duration and repetition rate of 80 MHz. For 2P excitation, the OPO is tuned to 1050 nm. To modulate the wavefront, we use a liquid crystal phase-only SLM (Meadowlarks HSP512L) which is imaged on the back focal plane of a water immersion objective (Zeiss W Plan-Apochromat 40x/1.0 DIC M27). After illuminating the sample, a second microscope objective (Zeiss EC Plan-NEOFLUAR 40x/1.3) images the excitation light pattern onto a CCD camera (Basler acA1300-30µm) and collects the fluorescence light which is steered towards a PMT (Hamamatsu H7422P-40) by a dichrocic mirror (Semrock Di03-R785-t1-25x36). Additional filters in front of the PMT (Thorlabs FESH0600, FESH0700) ensure that only nonlinear fluorescent light is detected.

For the scattering medium, a rough surface is introduced on a #1.5 cover slip by sandblasting its bottom surface with 220 grit sand. The beads (Fluospheres carboxylate, 1.0µm, red (580nm/605nm)) are drop-casted on the top surface of the coverslip, about 170µm above the scattering layer. They are covered by the NOA65 glue to create a near-index matched environment. Then, a second #1.5 cover slip is placed on top of the beads. Finally, the glue is cured under UV light, the curing process is kept short to avoid bleaching the beads. The fibres are provided by Thorlab lens tissue marked by a orange fluorescence pen, they are directly covered by a second #1.5 cover slip on top.

Reconstruction details

Algorithm 1: Multiplexed phase retrieval input : number of gradient descent, nn; step size, ∆; measurement matrix, A; measurement vector, y; column number, N obj . output: columns of transmission matrix,

t i (i = 1, 2 • • • N obj ). 1 for count ← 1 to nn do 2 t i = t i -∆ ∂f ∂t * i ∂f ∂t * i 2

end

In order to reconstruct t i in Eq.( 2) when p = 2. The Alg.1 is proposed to solve the following optimized problem:

min t i f, with f = y - N obj i=1 |At i | 4 2 2 , (4)
where N obj is chosen to be equal or larger than the actual number of sources behind the scattering medium. Some of the columns of the reconstructed T will be repeated if N obj is chosen larger (see Supplementary Information for more details). For the initialization, we randomly initialize each column t i of the transmission matrix. According to [START_REF] Remmert | Theory of complex functions[END_REF][START_REF] Fischer | Precoding and signal shaping for digital transmission[END_REF][START_REF] Bian | [END_REF], the derivative of the complex quadratic cost function with respect to t * i , i.e.

∂f ∂t * i = 4A † [( N obj i=1 |At i | 4 -y) |At i | 2 (At i )]. (5)
For the result in Fig. 2 In order to reconstruct T concatenated by vector t in Eq.( 3), the Alg.2 is proposed to solve the following optimized problem:

min t,o f, with f = Nm q=1 (y q -o T |F(a * q t)| 4 ) 2 , (6)
where a † q represents the qth row of A, y q represents the qth element of y. For the initialization, t is randomly initialized, and o is obtained by raster-scanning the object with a scanning device (applying phase ramps on the SLM in our case). The gradient ∂f ∂t * is derived as:

∂f ∂t * = 4 Nm q=1 (o T |F(a * q t)| 4 -y q )a q {F † [o |F(a * q t)| 2 (F(a * q t))]}. (7)
However, the auto-gradient derived by Pytorch are used in our code for faster computation speed. In each iteration, nn times of gradient descent are performed to reconstruct the t, then the o is updated by raster-scanning the sample with a scanning device and displaying compensate pattern arg(t * ) on the SLM simultaneously (applying a complete set of phase ramps on top of arg(t * ) on the SLM in our case). For the results in Fig. 3 The computational resource we used: GPU, 2 Nvidia GeForce RTX 2080 Ti; RAM, 64 GB.

u 1 u 2 = S t 1 l 1 t 2 l 2 (12)
satisfy Eq. (11). For example, (u 1 , u

2 ) = ( 1 √ 2 ( t 1 l 1 + t 2 l 2 ), 1 √ 2 ( t 1 l 1 -t 2 l 2 )
). Thus, the Eq.( 2) in the main text has infinite solution when p = 1.

In the p = 2 case, Eq.( 11) becomes

∀â † t 1 4 |â 1 | 4 + t 2 4 |â 2 | 4 = |â † u 1 | 4 + |â † u 2 | 4 . (13)
Let â1 = â2 = 0 and âi = ∀, i = 1 or 2, we have

u 1 =        α β 0 0 . . .        u 2 =        γ δ 0 0 . . .        . (14)
Let â1 = ∀, â2 = 0, we have

|α| 4 + |γ| 4 = t 1 4 . (15)
Let â1 = 0, â2 = ∀, we have

|β| 4 + |δ| 4 = t 2 4 . (16)
Let â1 = â2 = ∀, we have

|α + β| 4 + |γ + δ| 4 = t 1 4 + t 2 4 . ( 17
) Let â1 = -â 2 = ∀, we have |α -β| 4 + |γ -δ| 4 = t 1 4 + t 2 4 . (18)
Eq.( 17) plus Eq.( 18) and using Eq.( 15) and Eq.( 16) we have

(αβ * ) 2 + (α * β) 2 + 4|α * β| 2 + (γδ * ) 2 + (γ * δ) 2 + 4|γ * δ| 2 = 0. (19)

Transmission matrix based image reconstruction

Previous works that realized imaging beyond the memory effect (ME) by analysing the retrieved distance map between all the emitters (26,27,40,41), but they all used a camera to retrieve all the emission fluorescent patterns first, which is not sensitive enough for non-linear microscopy.

Here, we retrieved the relative position between every two emitters by using the transmission matrix T. Fig. 6 shows the image reconstruction work flow by analysing the T we retrieved in result Fig. 2e in the manuscript. If the two emitters are really close, the corresponding two columns in T should be highly correlated and provide relative position information. The relative position map of the two emitters is calculated by:

o i,k = |F 2d ( t * i |t * i | t k |t k | )| 2 , (23)
where F 2d represent the 2d Fourier transform, means element wise production. To infer whether the columns t i and t k are within or beyond the ME range, we study α = max{o i,k } max{o k,k } as a function of relative distance, where max{o i,k } stands for maximum value of o i,k . A given threshold, α tres of α is introduced to evaluate it. For example, if α is greater than α tres , the two columns belong to the same ME range. Otherwise, they belong to different ME ranges, the relative position map o i,k is reassigned to a zero matrix. The pairwise relative position map o i,k after thresholding is showing in Fig. 6.

Then the sub images of the object in the vicinity of the emitter k are obtained by O k = N obj i=1 o i,k . In order to merge the sub images to a global image, we also need the relative position vector r i,k of the two emitters i and k. If emitters i and k are in the same ME range, r i,k is a vector in the map o i,k that origin at the center and end at the narrow delta-like peak. If emitters i and k are beyond the ME range but emitter j is between them, we can still calculate the r i,k between them as r i,k = r i,j + r j,k . Therefore, even if the ensemble of emitters expands well 22 beyond the ME range, the full spatial distribution can be recovered if the different isoplanatic patches are "connected" by emitters.

Figure 1 :

 1 Figure 1: Experimental and reconstruction principles. (a) Simplified schematic view of a general fluorescence microscope with a single-pixel detector. Random wavefront generated by a SLM (each wavefront represents one row of matrix A) impinges on a scattering medium with a transmission matrix T and excites 2-photon fluorescence (2PF) of the extended object o. The fluorescence signal y is collected by a single-pixel detector (photo-multiplier tube (PMT)). Retrieval of the TM is achieved in two different extreme cases 1 and 2, where tailored algorithms are used for solving different models. (b) In case 1, T is assumed to have completely uncorrelated elements, i.e. a random matrix.We solve for T using as input the measurements y and the wavefront modulation embedded in the user-input matrix A. The reconstruction of T allows to focus on each guidestar in the object. (c) In case 2, T has correlations, i.e. the infinite ME range is considered, and is given by the pixel-wise product of a rank-1 matrix T and a "tilt matrix" F T , where F is a matrix corresponding to 2D Fourier transform. An iterative method is used where, in each iteration, we solve for T at first, then the wavefronts estimation are used for raster-scanning the sample to update o until convergence.

Focusing

Figure 2 :

 2 Figure 2: T retrieval beyond the ME. (a) 2PF signal detected (y) vs. number of random wavefronts. Representative wavefronts used are shown in the insets. The two red dashed line represents the average level of the first and last 100 signal samples, respectively. (b) Inspected focii (control CCD) for each column after retrieval of T. The insets show line profile. (c) Sum of all images for each focus demonstrating unique single focus for each column retrieved. (d) Brightfield image taken with control camera confirms the number of sources. (e) Imaging after analysis on reconstructed T using correlations between the different columns (The red circle in (c) depicts the memory effect range).

Figure 3 :

 3 Figure 3: T retrieval assuming ME correlations. (a) 2PF imaging by raster scanning the sample using the speckles depicted in (b), themselves captured with the CCD before each iteration. The images are low-resolution in order to speed up acquisition. High resolution images before (c1) and after two iterations (c2) show the cluster, and its morphology confirmed by the brightfield camera (c3).

Figure 4 :

 4 Figure 4: Retrieval of T for extended objects. The samples in (a) and (b) are fibres that coming from lens tissue marked by a fluorescence pen. The samples in (c) is a bunch of 1µm beads. (a1), (b1) and (c1) are the imaging result by scanning in 64 × 64 steps without wavefront correction. (a2), (b2) and (c2) are the corresponding results after the wavefront correction. Here, we use 16 random patterns scanning to generate the measurement matrix A, and put A and y as input for Alg.2. The samples in (a), (b) take 3 iterations to converge, and (c) takes 5 iterations. (a3), (b3) and (c3) are the corresponding bright field images.

and Fig. 4 , 8 ×

 48 N slm and 16 × N slm measurements are taken to get y respectively. They are obtained by using 8 and 16 random patterns on the SLM and raster-scanning the sample with them one by one, N slm = 16 × 16 measurements for each raster-scanning. The sampling strategy here can significantly reduce the sampling time if the galvo-mirror are used as scanning device and placed on the conjugate plane of the SLM. The higher resolution images in Fig.3and Fig.4are obtained by raster-scanning the sample with SLM in 64 × 64 steps. In each iteration, the reconstruction of t take less than 10 seconds on GPU.

Finally, we use

  the method proposed in(40) to merge all the sub images, and realize imaging beyond ME range. In short, the global reconstruction of O global can be obtained by composing all the sub images O k into one images, taking into account their relative positions with respect to the first emitter, r k,1 :O global = N obj k=1 O k (r -r k,1 ). (24)Therefore, a image beyond the ME range is obtained. The red dashed circle showing in the reference focusing image in Fig.6represents the ME range. Another result with 8 fluorescence beads is shown in Fig.7.

Figure 5 :Figure 6 :

 56 Figure 5: The fidelity of retrieval T with different N obj : Three different kind of random matrix T are generated for ground truth, they have 3 (red data), 7 (blue data) and 11 (green data) columns respectively. T reconstruction fidelity are evaluated when using different N obj . The each solid line represents the average of 10 realization, and the shadow represents the fidelity range distribution.

  ), (4) repeat (2) and (3) until a strong 2-photon signal is detected. (see methods for further details on the Algorithm 2). Since the matrix T has only one vector t as unknown, the multiplexed phase retrieval problem simplifies to a normal phase retrieval problem with only one column to be reconstructed. Hence, the measurement number decreases considerably going from N m ∝ N slm × N obj to typically N

3, we propose a synergistic algorithm to ensure convergence. As schematically shown in Fig. 1c: (1) collect measurement vector y with the measurement matrix A, and initialize o by raster-scanning the sample, (2) update t by solving Eq.(3) numerically, (3) update o by raster-scanning the sample with the compensate wavefront arg(t * m ∝ N slm . Note here that instead of only using an SLM to generate a measurement matrix A, we can use a scan element and SLM together to generate A: this allows to speed up measurements significantly (e.g. using galvanometric mirrors).

  , 16 × N obj × N slm random patterns are displayed on the SLM to obtain y, where N slm = 16 × 16 and N obj = 7. The reconstruction of Alg.1 takes less than 10 seconds Update o by raster-scanning the sample with the compensate wavefront arg(t

	on GPU.
	Algorithm 2: Phase retrieval + Scanning
		input : number of gradient descent, nn; step size, ∆; measurement matrix, A;
		measurement vector, y.
		output: aberration, t; object, o.
	1 while 2-photon signal is low do
	2	for count ← 1 to nn do
	3	t = t -∆	∂f ∂t * ∂f ∂t * 2
	4	end
	5

* ) 6 end

Acknowledgements L.V. acknowledges funding from the Swiss National Science Foundation (project P400P2_199329). S.Z. acknowledges funding from the China Scholarship Council (CSC) (202006280163). This project was funded by the European Union's Horizon 2020 research and innovation program under the FET-Open grant No. 863203 (Dynamic). The authors acknowledge the assistance of Eric Bezzam in optimizing the code.

Code available

Simulation codes are available at https://github.com/comediaLKB/Single_pixel_TM_recovery.

Competing interests

The authors declare no competing interests.

Supplementary Information:

The uniqueness of the solution depending on p To aid in understanding, we consider a scenario that only 2 beads under the scattering medium.

We aim to demonstrate that, given a sufficient number of random measurements, the solution to Eq.( 2) in the main text is non-unique when p = 1, while it is unique when p = 2. It's not hard to see that the conclusion can be generalized to the case of more than 2 beads.

The corresponding 2 columns of T ∈ C N slm ×2 are t 1 and t 2 , respectively. a † ∈ C 1×N slm represents one row of the measurement matrix A. and the corresponding fluorescence signal y is:

Assuming t † 1 t 2 = 0, which means the columns of transmission matrix are orthogonal to each other (since the transmission matrix are complex random matrix, this a weak assumption). We can then represent inner product in a canonical basis starting with

Eq.( 8) in this basis becomes:

where R represents a unitary basis rotation matrix, and âi represents the ith element of the vector â.

The question now is: Are there any (u 1 , u 2 ) = (l 1 , l 2 ) that satisfy the following formula?

Defining

into Eq.( 19), where i represents the imaginary unit and m, n, p, q are real, we have

Thus, m = n = p = q = 0. It means αβ * = γδ * = 0. So α or β = 0 and γ or δ = 0.

Therefore, we conclude that

Thus, Eq.( 2) in the main text has unique solution when p = 2.

T reconstruction quality for using different N obj

In order to estimate the T reconstruction quality using different N obj , three kind of random matrix T are generated for ground truth, the number of columns they contain is 3, 7 and 11 respectively. Different N obj ranging from 1 to 14 are using in Alg.1 to reconstruct T. The reconstruction fidelity is shown in Fig. 5, the red, blue and green are corresponding to the ground truth T with 3, 7 and 11 columns respectively.

The fidelity of each column t i is calculated between the reconstructed t i and every column of the ground-truth T then pick the maximum. The fidelity of the transmission matrix T is the average fidelity of all the column

As showing in the result, when the N obj is chosen smaller than the sources number, the reconstructed column t i is not guaranteed to match one of the column in the ground-truth matrix T. On the contrary, when the N obj is chosen bigger than the sources number, each reconstructed t i match one of the column in the ground-truth T. Actually, in this case some of the columns of the reconstructed T will be repeated. In the simulation, N m = 65536 and N slm = 16 × 16. The each solid line represents the average of 10 realization, and the shadow represents the fidelity range distribution.


Armelle Brun
email: armelle.brun@loria.fr

Sylvain Castagnos
email: sylvain.castagnos@epfl.ch

Anne Boyer
email: anne.boyer@loria.fr

A positively directed mutual information measure for collaborative filtering

Keywords: Mutual Information, Neighbor selection, Item-based approach

As resource spaces become ever larger, the need for tools to help users find pertinent and reliable resources quickly and easily is more and more acute. Recommender systems are an efficient way to tackle the problem of information overload, as they enable to inference from users' past behavior to suggest resources the users have not seen yet. Collaborative filtering, that uses ratings of user on items, has become a very popular technique for recommender systems. One crucial step of item-based collaborative filtering is the computation of pair-wise similarity between items. In order to improve the accuracy of collaborative filtering, we propose a new approach for the computation of similarities, by using mutual information. In this work, only positive and negative evidences are considered for training, whereas classical approaches use ratings. During the prediction step, we do not use the classical K value (KNN neighbors) as the size of the neighborhood of each item, but a size value that depends on the considered item. The proposed method is evaluated on the well-known MovieLens dataset. Our experiments show an improvement in the accuracy of collaborative filtering.

Introduction

Internet makes available to online users an over-abundance of information, e-services and products. Users have to decide which information to consult, which service to use or which product to buy. Recommender systems [START_REF] Adomavicius | Toward the next generation of recommender systems : A survey of the state-of-the-art and possible extensions[END_REF] aim at finding, given a user, the adequate items from the huge number of available items. An item is either an information, or an on-line service or a product, and is the elementary unit we consider.

Collaborative Filtering (CF) is a popular recommendation technique, which uses similarities between users (user-based approaches) or items (item-based approaches) to predict the most pertinent items for a given user. The input of the system is a set of ratings of users on a set of items, and the goal is to predict how well a user will like an item that he/she has not rated given his/her past ratings and the ratings of other users [START_REF] Resnik | Grouplens: An open architecture for collaborative filtering of netnews[END_REF][START_REF] Shardanand | Social information filtering: Algorithms for automating "word of mouth[END_REF][START_REF] Breese | Empirical analysis of predictive algorithms for collaborative filtering[END_REF][START_REF] Sarwar | Item-based collaborative filtering recommendation algorithms[END_REF]. Ratings can be explicitely given by the users or implicitely derived from the analysis of the users' actions, in the available logs or by mining usage [START_REF] Chan | A non-invasive learning approach to building web user profiles[END_REF].

This paper focuses on the item-based approach, that associates to each item its set of nearest neighbors according to a similarity measure. The prediction of a user's rating on an item is based on all the items he/she has already rated, or the nearest neighbors (mentors) of this item.

Similarity between two items represents how these two items are similarly co-voted. We propose a new way to compute item-to-item similarities, relying on mutual information [START_REF] Shannon | A mathematical theory of communication[END_REF]. Mutual information (MI) is an information-theoric measure of the dependence of a variable on another one. Furthermore, we propose to compute item similarities when only positive or negative evidences are available. The advantage of this method is the reduction of the rating scale to only two states : "I like" or "I dislike". It makes easier the task of the users who do not have anymore to rate items but only to express a global opinion. Ratings that correspond to neutral opinions are considered a non-informative and are discarded.

In order to decide which items to use during the prediction phase, and therefore reduce time complexity while guarantying a high level of accuracy, we propose to use an original way to select neighborhood, that is an adjustable neighborhood depending on the item and its reliability, instead of the K nearest neighbors approach as introduced in [START_REF] Resnik | Grouplens: An open architecture for collaborative filtering of netnews[END_REF].

The rest of this paper is organized as follows : section 2 describes the item-based approach. Similarity measure based on mutual information is first described in section 3, then a new similarity measure based on mutual information is proposed. Section 4 details an original way to determine the optimal size of the neighborhood of a given item. The experimental dataset and protocol are described in section 5. Performance of the propositions presented in section 3 and 4 are assessed and analyzed in section 6. Section 7 draws the conclusion and section 8 presents future work.

ITEM-BASED COLLABORATIVE FILTERING

Item-based collaborative filtering algorithms explore the relationships between items, and then compute recommendations for the active user by finding items similar to the items he/she liked. The item-based approach is well suited to applications where the set of items is relatively static [START_REF] Castagnos | Personalized communities in a distributed recommender system[END_REF].

Let U be the set of N users, a ∈ U the active user and I the set of M items with t ∈ I the target item. r ui corresponds to the rating of user u ∈ U on item i ∈ I. Let sim(i, j) be the similarity value between the two items i and j. p at is the prediction of rating of the active user a on the target item t.

The procedure of predicting ratings using the similarity measure can be described as follows:

1. compute the similarity sim(i, j) ∀(i, j) ∈ I × I.

2. select the neighborhood of each item.

3. compute the prediction of rating p at for the targeted item t and the active user a.

Computing Similarity Between Items

One crucial step in an item-based algorithm is to compute the similarity between items. The basic idea in similarity computation between two different items i and j, is to determine the set of users who rated both items and then to apply a similarity computation technique. An overview of the most usual similarity measures can be found in [START_REF] Spertus | Evaluating similarity measures: A large-scale study in the orkut social network[END_REF][START_REF] Candillier | Comparing state-of-the-art collaborative filtering systems[END_REF], we only present here the most popular: the cosine and the Pearson coefficient measures.

Cosine Measure Item-based approach considers items as vectors in user space whose components are the users ratings. The cosine similarity between items i and j is computed as the cosine of the angle between -→ i and -→ j as follows:

sim(i, j) = cos( -→ i , -→ j ) = -→ i • -→ j || -→ i || || -→ j || (1)
The drawback of cosine measure is that the difference in rating scale between users is not taken into account [START_REF] Sarwar | Item-based collaborative filtering recommendation algorithms[END_REF].

Pearson Correlation Coefficient

The Pearson correlation coefficient evaluates the deviation from the mean rating of items.

sim(i, j) = P CC(i, j) = k (r ki -r i )(r kj -r j ) k (r ki -r i ) 2 k (r kj -r j ) 2 (2)
Where r i is the mean rating of item i and k sums over users that have both rated item i and j.

The advantage of Pearson coefficient is that both negative and positive correlation values are obtained. It has been shown in [START_REF] Breese | Empirical analysis of predictive algorithms for collaborative filtering[END_REF][START_REF] Candillier | Comparing state-of-the-art collaborative filtering systems[END_REF] that the Pearson Correlation Coefficient (PCC) leads to a higher accuracy than cosine measure. An additional measure can also be used, the adjusted cosine [START_REF] Sarwar | Item-based collaborative filtering recommendation algorithms[END_REF] that corresponds to the cosine of items deviation from the user mean rating. Adjusted cosine is similar to Pearson and offsets the cosine drawback by subtracting the corresponding user average for each covoted pair.

Selecting the Neighborhood of Items

To compute p at , similarities between the target item t and all other items are used. However, to speed up the prediction computation, only a subset of these items is used. Usually, most similar items to t are used (called nearest neighbors). A value of K is fixed a priori and K items neighbors are considered [START_REF] Goldberg | Eigenstate: A constant time collaborative filtering algorithm[END_REF][START_REF] Sarwar | Item-based collaborative filtering recommendation algorithms[END_REF][START_REF] Candillier | Comparing state-of-the-art collaborative filtering systems[END_REF] to compute predictions.

Predicting

In item-based collaborative filtering algorithms, the prediction p at is computed on the items (and the corresponding ratings) that the user a has already voted. These items are weighted according to their correlations with item t. An overview of the prediction methods is presented in [START_REF] Candillier | Comparing state-of-the-art collaborative filtering systems[END_REF].

The weighted sum, that is often used in item-based collaborative filtering, evaluates p at by computing the weighted sum of the ratings given by user a on the neighbor items of t.

p at = i sim(i, t) * r ai i |sim(i, t)| (3)
where i sums over all items in the nearest neighbors of item t, that have been rated by the active user a.

The following prediction method is classically used in the user-based algorithms [START_REF] Yu | Selecting relevant instances for efficient and accurate collaborative filtering[END_REF], less often in item-based approaches [START_REF] Lee | Improvements to collaborative filtering systems[END_REF]. The computation of p at considers the sum of the target item mean rating and the weighted sum of deviations from the mean rating of items that have been rated by user a.

p at = r t + i sim(i, t) * (r ai -r i ) i |sim(i, t)| (4)
where r i represents the mean rating of item i. This method will be used in this article to compute predictions, since it gets better results.

Assessing Performance

Performance of a recommender system can be evaluated in several ways, we are interested here in accuracy and coverage rate [START_REF] Papagelis | Qualitative analysis of user-based and item-based prediction algorithms for recommendation agents[END_REF].

The accuracy of a recommender system can be evaluated by using various metrics [START_REF] Herlocker | An algorithmic framework for performing collaborative filtering[END_REF]. One of the most widely used is the Mean Absolute Error (MAE). MAE measures the deviation of the recommendations from their true user-specified values. The lower the MAE is, the more accurate are the predictions of the recommender system. Given a test corpus T , composed of triplets (u, i, r) where u is a user, i an item and r the rating u gives on i, MAE is computed as follows:

M AE = 1 |T | (u,i,r)∈T |p ui -r| (5)
Accuracy of a recommender can also be evaluated by using the HMAE (High MAE) measure [START_REF] Baltrunas | Dynamic item weighting and selection for collaborative filtering[END_REF]. HMAE represents the MAE computed only on the highest rating values. Two HMAE values can be considered. The first one, we call the TestHMAE, computes the accuracy on the high ratings of the test corpus. The TestHMAE represents the accuracy on the items the users like, it is related to the recall measure. The second measure, we call the RecHMAE, computes the accuracy on the items the recommender predicts with high ratings values. The RecHMAE is related to the precision measure. Both RecHMAE and TestHMAE are important to compute the accuracy of a recommender. Indeed, the items the recommender proposes to the user (items he rated highly) must be items the user actually likes, and conversely, the items a user likes have to be proposed by the recommender.

Coverage rate is used in addition to MAE and HMAE measures. Coverage computes the percentage of items for which a recommender can provide a prediction. In the case of items that have not been sufficiently co-voted, no reliable similarity values are computed, thus no prediction can be performed. In this article, coverage is studied, as some prediction methods may not reach a 100% coverage rate.

MUTUAL INFORMATION-BASED SIMILARITY MEASURE

To improve the accuracy of a recommender system, we propose in this article to use mutual information as a basis to compute the similarity measure between two items. Mutual information has already been introduced in user-based collaborative filtering algorithms. To our knowledge, it has not been much used to compute dependency between items [START_REF] Yu | Selecting relevant instances for efficient and accurate collaborative filtering[END_REF].

In [START_REF] Baltrunas | Dynamic item weighting and selection for collaborative filtering[END_REF][START_REF] Yu | Feature weighting and instance selection for collaborative filtering: An information-theoretic approach[END_REF][START_REF] Ziqiang | Collaborative filtering based on mutual information[END_REF], the similarity between two users is computed on co-voted items with the Pearson Correlation Coefficient. A specific weight is then assigned to each co-voted item, this coefficient is defined with the mutual information measure. Mutual information can also be used during the prediction phase [START_REF] Ziqiang | Collaborative filtering based on mutual information[END_REF]. It allows to rank the relevance of the active user to the target item, thus performing feature selection.

In information theory, mutual information represents a measure of statistical dependence between two variables X and Y with associated probability distributions p(x) and p(y). In our context, x and y are rating values.

Following Shannon [START_REF] Shannon | A mathematical theory of communication[END_REF], the mutual information (MI) is defined as:

M I(X; Y ) = x y p(x, y) log p(x, y) p(x)p(y) (6)
where x (respectively y) sums over values (ratings) of X (respectively Y ).

In this work, we use the normalised mutual information (NMI) [START_REF] Fred | Robust data clustering[END_REF], thus weights are in the range [0; 1]. NMI is defined as follows:

N M I(X; Y ) = 2 • M I(X; Y ) H(X) + H(Y ) (7)
where H(X) is the entropy of X and is computed as:

H(X) = - x p(x) log p(x) (8)
From now on, we will refer to normalized mutual information as mutual information.

A large mutual information value between the votes of two items reflects a high level of dependence. However, this dependence can be of two kinds. On the one hand, it can be positive: the two items are mainly rated with similar values. On the other hand, the dependence can be negative: the two items are opposed items, they are rated with opposite values -when one item is liked by a given user, the second one is usually disliked. Thus mutual information does not give any information about the nature of this dependence: a high value may represent a positive or a negative dependence.

In the framework of recommender systems, mutual information is used to compute similarity between items, then to predict the rating of the target item. The nature of the dependence is thus a crucial information, that can be used to improve the prediction capability of the recommender system.

Using binary ratings

We propose, as mentioned in [START_REF] Herlocker | Evaluating collaborative filtering recommender systems[END_REF], to group the range of ratings into two sets: positive opinions and negative opinions. For example, on the most commonly used MovieLens dataset, ratings range from 1 to 5, where 1 represents the lowest possible rating and 5 represents the highest.

We propose to merge ratings 1 and 2, representing the evidence "do not like" and merge ratings 4 and 5, to represent the evidence "like". We consider that rating 3, that represents a neutral opinion, is a non-informative rating. It is the reason why we propose to discard it from the computation of the mutual information on the training dataset, as it can be considered as noise, thus decreasing performance.

In the case of binary opinions, Equation ( 7) becomes:

N M I(i; j) = 2 H(i)+H(j) • ( p(i l , j l ) log p(i l ,j l ) p(i l )p(j l ) (A) + p(i l , j d ) log p(i l ,j d ) p(i l )p(j d ) (B) + p(i d , j l ) log p(i d ,j l ) p(i d )p(j l ) (C) + p(i d , j d ) log p(i d ,j d ) p(i d )p(j d ) ) (D) (9)
where i l represents a positive assessment (liked the item) and i d a negative assessment (disliked the item). p(i l , j l ) represents the probability, for a given user, of liking both items i and j. This probability is computed as the normalized counts. p(i l ) represents the probability of liking item i. This measure will be referred as the Global Mutual Information (GMI).

Positive Mutual Information Measure

As presented previously, we are interested in using a similarity measure that integrates the nature of the dependence between two items. In this way, we split the mutual information into two sub-measures. The first one reflects the "positive" dependence, i.e. how much items have similar ratings. This measure contains the terms (A) and (D). Indeed, a high value of term (A) represents that most users who liked i also liked j and a high value of term (D) represents that users who did do not like item i did not like item j neither. The second sub-measure represents "negative" dependence, i.e. how much items have opposite ratings. In this article, we are interested in the first measure, that is defined as:

P M I(i, j) = 2 H(i)+H(j) • ( p(i l , j l ) log p(i l ,j l ) p(i l )p(j l ) + p(i d , j d ) log p(i d ,j d ) p(i d )p(j d ) ) (10)
This measure will be refered to as the Positive Mutual Information (PMI) as it represents to what extent two items are similarly rated. This measure can have both positive and negative values.

For comparison purposes, we also study the measure that contains (B) and (C) terms, that will be refered to as Negative Mutual Information measure (NMI).

We can notice that mutual information is highly influenced by the quantity of observed data. The smaller the size of observable data is, the less reliable the MI will be. That is why similarity between two items is computed only if a significant number of users have rated both items. In this paper, a threshold of 5 has been defined (see section 5.2).

Once similarities between items are evaluated, the ratings can be predicted. This prediction can be either computed by using all items or using a subset of items: its nearest neighbors.

SELECTION OF THE NEIGHBORHOOD FOR A GIVEN ITEM

In this section we address the problem of deciding which items to use during prediction and try to remove the unreliable ones to improve the quality and scalability of collaborative filtering.

The key idea is to highlight the fact that, weak dependent items (with low similarity value) are low informative for prediction. The use of such items may decrease the quality of the prediction. Furthermore, generating predictions over relevant items instead of operating over the entire dataset may decrease complexity time.

The most popular way to decide which items to use for prediction is to fix an a priori value K, that represents the number of items to use. The items are the K nearest neighbors of item t, i.e. items with the highest values [START_REF] Karypis | Evaluation of item-based top-n recommendation algorithms[END_REF][START_REF] Baltrunas | Dynamic item weighting and selection for collaborative filtering[END_REF][START_REF] Candillier | Comparing state-of-the-art collaborative filtering systems[END_REF]. The value of K is set experimentally and accuracy of collaborative filtering is influenced by the size of neighborhood.

The drawback of the K nearest neighbors method is that exactly K neighbors are used to compute ratings. Can the optimal predictions be computed by always using K neighbors ?

We propose here to improve this K nearest neighbor approach: the size of the neighborhood is now dependent on the considered item. Indeed some items may have few number of highly similar items, this small set may be sufficient to compute prediction. Conversely, some items may not have highly similar neighbors, and a higher number of neighbors may be used to compute predictions. Given an item, we propose to define its neighborhood as the set of similar items so that the value of their similarities sums up to a fixed value noted V . This approach has the advantage to determine the adequate size of the neighborhood for each item.

EXPERIMENTAL EVALUATION

The MovieLens Dataset

In our experiments we use the MovieLens dataset3 , containing 100K explicit ratings, for 1682 movies (items) and 943 users. Each rating varies in the range 1 to 5. Data sparsity on this corpus is 96%. This corpus is divided into five parts; to perform five-fold cross-validations, training is made up of 4 parts and testing is the last part.

As we consider that ratings with a value of 3 correspond to a neutral opinion, we propose, in a first step, to discard them from training corpus ; they represent 27% of the dataset. Obviously, all ratings in the test set are considered. The resulting data set is made up of 24% of negative opinions (merging ratings 1 and 2) and 76% of positive opinions (merging ratings 4 and 5).

Experimental Protocol

As previously mentioned, mutual information is unreliable when only few data are available. Therefore, we introduce a threshold -which represents the minimal amount of required data -in order to compute a reliable mutual information value. Otherwise, using a MI value when the threshold has not been reached may introduce a bias in the model. In that case, a 0 value is set in the similarity matrix, meaning that items are independent. A similar threshold is used for other methods.

RecHMAE and TestHMAE are evaluated on high ratings, we consider here that high ratings correspond to the "like" evidence, thus ratings 4 and 5.

RESULTS AND DISCUSSION

Comparison of Similarity Measures

In a first step, we implement various similarity measures: the cosine measure (Equation ( 1)) and the Pearson coefficient (Equation ( 2)) which are broadly used in the state of the art. In order to compare measures proposed in this article the GMI (Equation ( 8)) and PMI similarity measures (Equation ( 9)) are also implemented. For comparison purposes, NMI is also presented.

Recall that PMI is a sum of two terms:

the first one, called the "like term" (LT), -2 H(i)+H(j) p(i l , j l ) log p(i l ,j l ) p(i l )p(j l ) , -the second one, called the "dislike term" (DT), -2

H(i)+H(j) p(i d , j d ) log p(i d ,j d ) p(i d )p(j d ) .
In many research areas, the "like term" is often used alone, as a simplified version of the mutual information.

We study here the influence of each subterm on the MAE, in order to determine the pertinence exploiting both subterms.

Note that, before evaluating the MAE or HMAE scores, all predicted ratings are rounded as the ratings given by the users are integer values.

We can remark that for all similarity values, the similarity matrix contains about 77% of null values. This rate is explained by the fact that the similarity value between two items is set to 0 if they have not been sufficiently co-voted. A 0 value can also mean that two items are independent. When a no minimum co-voted value is set, the similarity matrix contains only about 45% null values.

Table 1 presents MAE and the coverage rate (defined as the percentage of predicted items), on the five test corpora, using all neighbors, for all the similarity measures listed above. We can first notice that all the methods have quite similar coverage rates. A 100% coverage rate is never achieved, this can be explained by the fact that a subset of pairs of items in the dataset does not reach the covoted threshold (see Section 2.4). We can also notice that, although ratings with a value 3 have been discarded, the coverage rate is high.

The Pearson Coefficient results in a better MAE than the cosine measure (2%), as previously shown in [START_REF] Candillier | Comparing state-of-the-art collaborative filtering systems[END_REF]. The Global Mutual Information measure performs slightly worse than Pearson, but performs better than Cosine. When comparing measures using mutual information, we can notice that the Positive Mutual Information, that uses a subset of terms of the GMI, slightly improves its MAE off 1.4%. Moreover, NMI, that represents how opposed items are, reaches the worst MAE value among all similarity values. We can deduce that the two sub-terms (B) and (C) from the GMI, that are not part of the PMI, are not good terms to compute the similarity. Indeed, discarding these two terms does not increase the MAE, it does even decrease it.

In order to study the pertinence of using both terms in PMI, we study the MAE obtained for the LT or DT. Both terms lead to higher MAE (respectively 0.8% and 1.4% higher than PMI), which shows that the combination of both subterms reinforces the accuracy of the prediction.

We can notice that NT leads to a higher MAE value and lower coverage rate. This can be explained by the small percentage of disliked items within the dataset (24%), which lowers the quality of the model.

We are also interested in the study of TestHMAE and RecHMAE values for all similarity measures, presented in Table 2. The coverage rate for TestHMAE is constant as it represented the rate of items highly rated in the test corpus. Concerning RecMAE, all methods have comparable coverage rates: they all rate about 25.4 % of the item with a high value. When comparing Pearson and GMI, we can remark that they have opposite behaviors: Pearson has a better TestHMAE whereas GMI has a better RecHMAE. PMI, that leads to the best MAE, slightly improves TestHMAE and RecHMAE over all similarity measures.

K nearest neighbors

In this section, we study the variation of the MAE according to the size of the neighborhood used to compute the prediction. The neighbors used are the most correlated neighbors. A value of K is fixed a priori and a rating is computed according to K neighbors. We can notice that whatever the value of K is, the coverage does not vary. We can notice that, as presented in [START_REF] Goldberg | Eigenstate: A constant time collaborative filtering algorithm[END_REF][START_REF] Sarwar | Item-based collaborative filtering recommendation algorithms[END_REF][START_REF] Liu | An optimized collaborative filtering approach combining with item-based prediction[END_REF], MAE does not highly improve when using only a subset of the nearest neighbors. However, the improvement of PMI and GMI is five times higher than the improvement of Pearson. We can also notice that the optimal value of neighbors for Pearson (about 70) is higher than the optimal value for PMI and GMI (about 30).

A dynamic size of the neighborhood

When using the K nearest neighbors presented in the previous section, exactly K items are used to compute prediction. We can suppose that some ratings may be computed by using less than K neighbors (for example when a few number of highly correlated neighbors exists). In some cases, more than K neighbors may be useful to accurately compute one rating (when only low correlated items are available). In this section we present an original way to choose neighborhood: it is now made up of the highest similar items such that the sum of the values of their similarities is over a value V . This approach has the advantage of selecting a neighborhood whose size depends on the target item. For example, when highly correlated items are available, a few numbers of neighbors is used, and low-correlared neighbors are not considered, avoiding a decrease of performance. As for K nearest neighbors, this way to select neighborhood leads to a constant coverage. Figure 2 shows the evolution of the MAE for Pearson coefficient, according to the value V . We can notice that the lower MAE is obtained with a value V = 25. This MAE is similar to the one reached when using a fixed number of neighbors K (section 6.2). A value V = 25 corresponds to 56 neighbors on average. The optimal number of neighbors with the KNN method was 70, the number of neighbors to use is thus reduced by 20%.

Figure 3 represents the evolution of MAE for PMI and GMI according to the V value. We can remark that this new method slightly improves PMI and GMI. The value of the MAE obtained for GMI is 0.689 whereas the one obtained with KNN was over 0.691. For PMI, MAE obtained is also improved compared to KNN. Concerning the value V , the optimal value for PMI is 4.5, that corresponds to on average 24 neighbors (23 % less than when using KNN). The maximum number of neighbors used is 97. To conclude, the use of the threshold V improves the K nearest neighbors while reducing the mean size of the neighborhood.

Influence of the Similarity Threshold

In this section, we are interested, as in the two previous sections, in a better way to select the neighbors to compute prediction. We study the influence of an item, according to its similarity with target item t. We introduce here a similarity threshold T that represents the lower bound of pertinence of a similarity value: items correlated with a value below T are not taken into account.

We consider that low dependent items are not useful for prediction, and may lead to even lower performance. The question is which value of T should be chosen for threshold ? In Table 3, MAE and RecHMAE are presented according to the threshold T , for several similarity measures. On the one hand, we can notice that the coverage rate decreases quickly with the threshold value: this is due to the fact that only few similarity values are above the similarity threshold when it grows up. For example, when the threshold is fixed at a value of 0.3 the coverage rate dramatically decreases by 85%. We can also notice that decrease in the coverage for Pearson and Cosine is slower than for GMI and PMI. For example, for the same threshold of 0.3, Pearson computes a prediction for 97.9% of the test set.

On the other hand, we can see that the MAE has a similar variation: accuracy is improved with the threshold value. For Cosine, MAE slowly decrease according to T . We can notice that with low values of T , the MAE of GMI and PMI increases, while the coverage decreases slightly. For example, for GMI with a threshold T = 0.2, MAE is increased by 6% compared to the use of all neighbors. This can be explained by the low number of neighbor items used to compute prediction: when using a threshold T = 0.2, the number of neighbors used is on average 30 times lower than when using a threshold T = 0.0. The difficulty is to find the right balance between the MAE and the coverage rate by tuning this threshold experimentally. However, Pearson MAE does not decrease when the threshold increases, it does even increase. When the threshold is fixed to 1.0, coverage is equal to 0.6% and MAE is 0.872: when only the items correlated with a value of 1.0 are used (the highest correlation value that can be reached), MAE increases by 20% compared to using all positive correlations. A similar behavior is observed when 5 ratings are used (without binary ratings). Cosine, GMI and PMI have expected behaviors: MAE decreases when the value of the threshold is high. For example when the threshold is fixed to 0.5, the MAE of PMI is equal to 0.423. The most correlated the items used for prediction are, the most accurate the recommender is. Of course, coverage highly decreases according to the threshold. These results can be linked with Figure 1: when Pearson is used, a low improvement of the MAE is reached when using KNN, whereas a large improvement is reached for PMI and GMI.

We can also notice that when a threshold is fixed, GMI is more accurate than GMI, whereas PMI is more accurate when all items are used.

MAE does not get better when using a high value of T for Pearson, since a high Pearson value does not reflect that the two items have identical ratings: a correlation of almost 1 can be obtained even when items are not identically rated: this can for example be due to the influence of the deviation to the mean.

To highlight the bad influence of weakly dependent items in the computation of prediction, we studied the MAE of PMI on the 0.3% of the corpus that correspond to a 0.237 MAE value (threshold value T = 0.6). Computing the MAE on this corpus without any threshold (using all neighbors) leads to a MAE of 0.57. It shows that using a large number of weak dependent items actually increases MAE drastically. Similar results are obtained with GMI.

Combining several similarity measures

In the previous sections, we first presented a new method that selects the neighbors to use for prediction so that the sum of their similarities is equal to a fixed value V . An improvement of the MAE of PMI and GMI has been reached compared to KNN while having a constant coverage. We then studied MAE when only highly similar neighbors (with a similarity value higher than a fixed threshold T ) are used for prediction. We noticed that this method reduced dramatically the coverage. However, MAE of PMI and GMI highly decreased according to this threshold.

In this section, we aim at taking advantage of both preceding methods, by combining them: when neighbors with similarity higher than a fixed threshold are available, only these neighbors are used (section 6.4), when no such neighbors are available, neighborhood with dynamic size is used (section 6.3). In Table 3, we have shown that GMI was the most accurate measure when threshold grew, whereas it was less accurate when all neighbors were used. That is why, we used GMI when neighbors with similarity higher than threshold T are available, and PMI is used in other cases. 4 presents the evolution of the MAE according to T . We can remark that the evolution of the MAE according to the threshold T has the same aspect than the MAE for GMI used alone: it first increases, then decreases. In this combination, coverage is constant. The MAE obtained by using all neighbors with GMI used alone was 0.696 and 0.686 for PMI. The combination of both measures reaches a MAE of 0.677 with a threshold value of 0.4 that improves the reference MAE by 2.7%.

CONCLUSION

This paper aims at improving accuracy of item-based collaborative filtering approach and presents a new way to compute the similarity between items. This measure, called Positive Mutual Information (PMI), relies on the information theory measure of mutual information. It has the characteristics of using only a subpart of the classical mutual information measure. We compute its performance on the well-known MovieLens dataset and compare it with other well-known similarity measures such as the Cosine, Pearson correlation and the General Mutual Information. We also propose to consider only two assessment states instead of the usual rating scale. The users only have to express their opinions in terms of "I like it" or "I dislike it". All the "no opinion" values are discarded. Such ratings are easier to estimate from logs or usage, that is to say the implicit ratings that users should put on items if they had explicitely voted those items.

We show that the PMI increases the performance of both Cosine and Pearson correlation when used in identical contexts. It also increases performance of the General Mutual Information. This result is highly important as it shows that the computation of similarities between items can be restricted to those with similar behaviors, while decreasing time complexity.

We also propose an original way to compute the neighborhood of a given item. This neighborhood is made up of the most correlated items so that the sum of their similarities sums up to a fixed value. We showed that it decreases the MAE while using a smaller neighborhood size on average.

Finally, we showed that MAE can be increased by combining both GMI and PMI, leading to a MAE of 0.677 that corresponds to an improvement of 2.7%

Our major contributions are mainly the computation of similarity based on mutual information, the improvement of the accuracy when using only two opinions from the users ("I like" or "I dislike") and an item-dependent neighboorhood size. The basic hypothesis is that it is better for a system not to recommend any item than to suggest an item the user will dislike. Considering the HMAE, this goal has been achieved.

PERSPECTIVES

In a near future we plan to investigate the impact of the use of opposite items (terms (B) and (C) of Equation 7). We plan to replace the GMI equation by a combination of two sub-equations depending on the item characteristics. For example, if items have similar behaviors (as studied in this paper) the PMI will be used, otherwise NMI will be used.

We also plan to associate a confidence value with each similarity measure. This confidence value may be related to the number of users involved in the computation of the similarity. This value will be used to weight the similarity measure during the prediction phase.

Fig. 1 .

 1 Fig. 1. Evolution of the MAE value according to the number of neighbors used to compute prediction

Fig. 2 .

 2 Fig. 2. Evolution of the MAE value according to the value of the threshold neighborhood for Pearson

Fig. 3 .

 3 Fig. 3. Evolution of the MAE value according to the value of the threshold neighborhood for GMI and PMI

Fig. 4 .

 4 Fig. 4. Evolution of the MAE value according to the value of the threshold neighborhood for GMI and PMI

Table 1 .

 1 Comparing similarity measures

	measure MAE coverage
	Cosine 0.0.699 98.4 %
	Pearson 0.693 98.5 %
	GMI 0.696 98.5 %
	PMI 0.686 98.4 %
	NMI 0.724 99.3 %
	LT	0.692 98.5 %
	DT	0.696 94.8 %

Table 2 .

 2 HMAE values for all similarity measures

	measure Test	cov.	Rec	cov.
		HMAE		HMAE
	Cosine 0.613	54.9 % 0.608 24.7 %
	Pearson 0.603	54.9%	0.606	25.6 %
	GMI 0.608	54.9%	0.603	25.2 %
	PMI 0.596 54.9 %	0.598 25.4 %
	LT	0.608	54.9%	0.603	25.2 %
	DT	0.601	54.9%	0.603	25.5 %

Table 3 .

 3 Influence of the similarity threshold T on MAE and coverage for Pearson, Cosine, GMI and PMI

	Threshold Pearson Pearson Cosine Cosine GMI	GMI PMI	PMI
		MAE	coverage MAE coverage MAE coverage MAE coverage
	0	0.693	98.5 % 0.699	98.4	0.696 98.5 % 0.686 98.4 %
	0.1	0.693	98.3 % 0.696	98.5 % 0.705 93.4 % 0.685 97.6 %
	0.2	0.693	98.2 % 0.686	97.1 % 0.737 46.1 % 0.700 90.2 %
	0.3	0.700	97.9 % 0.684	88.9 % 0.610 8.4 % 0.735 56.1 %
	0.4	0.717	95.5 % 0.689	66.7 % 0.402 1.9 % 0.675 10.2 %
	0.5	0.749	88.9 % 0.681	35.6 % 0.365	0.9 % 0.423 1.4 %
	0.6	0.788	76.2 % 0.663	9.8 % 0.237	0.3 % 0.237	0.3 %
	0.7	0.832	59.0 % 0.504	1.3 % 0.151	0.1 % 0.151	0.1 %
	0.8	0.880	37.6 % -	-	-	-	-	-
	0.9	0.914	14.0 % -	-	-	-	-	-
	1.0	0.872	0.6 % -	-	-	-	-	-

http://www.movielens.org


Rhetoric and other language sciences in Syriac 1

Margherita Farina -Mara Nicosia

The pedagogical and epistemological role of language sciences

Rhetoric and grammar constituted the basics of the Syriac cursus studiorum (course of study). For centuries, throughout Late Antiquity and the Middle Ages, education in language sciences in Syriac ecclesiastical culture was considered one of the primary instruments to access the Scriptures and, ultimately, Christian theology. Moreover, within a more general cultural perspective, grammar, rhetoric and dialectic constituted a progressive pathway to knowledge. Language sciences were meant to provide the student with the analytical method, syllogistic reasoning (Syr. sulogismā), governing all other disciplines, from astronomy and geometry, to medicine and physics.

Syriac Christianity first developed in the second century in the city of Edessa (Syr. ʾUrhay). Between the third and sixth centuries, it spread westwards throughout Syria-Palestine and eastwards through Mesopotamia, Iran, India and Central Asia. Through the translation of the Old and New Testaments, between the second and third centuries, the Aramaic dialect of Edessa became the religious, liturgical and cultural language of eastern Christianity, under the name of Syriac (lešānā suryāyā). Refounded as a Macedonian colony in 304 BCE by the diadochus Seleucos I Nicator, Edessa was a Hellenistic polis, with Greek institutions, and was led by a bilingual Greco-Aramaic cultural élite. Several middle-eastern cities, like Edessa, participated in the Greco-Roman system of education, which was devised for Greek-speaking people, but was progressively adapted into Aramaic, through a process that lasted several centuries and that reached an apex between the fifth and the seventh centuries CE. At the same time, the centres of learning for Christian society in these cities shifted from Greek schools to Christian schools belonging to different communities (such as those of the Persians and of the Armenians, in Edessa in the fourth-fifth centuries). Subsequently monasteries also took a central role in learning, among which one of the most famous was Qenneshre "The eagle's nest", on the Euphrates, east of Aleppo.

These different institutions inherited the Hellenistic system of education, organised into three stages: primary, propaedeutic, and advanced. 2 We can learn about the Syriac school curriculum from the accounts of the organization of the School of Nisibis (founded in the late fifth century CE), a prominent center of learning. The primary teaching focused on basic literacy, on recognizing the letters of the alphabet and on mastering basic reading. In Syriac education, this teaching, as described in a number of literary sources, was provided by the maqryānā, "reader" (from the root qrʾ "to speak out loud", also "to read"). While being initiated into literacy, pupils also received the foundations of religious education. Hence, they first read the Psalms (which were learnt by heart through chanting), then the Gospel and the Pauline Epistles, and finally the Old Testament. The maqryānā also taught basic notions of grammar. A certain level of primary education was imparted by the mhaggyānā, "the one who makes spelling", but also "the one who makes meditating" (from the root hgy "to think, meditate"), who was in charge of teaching how to spell words. The method of this basic teaching is described in the sixth-century text Cause of the foundation of schools: the simple letters (ʾathwatā pšiṭātā) are read out loud to the child, who repeats them. Then the letters are connected in order to form nouns, which the child has to spell (nehgē) and practice. Due to Syriac consonantal writing, spelling and vocalizing a text implies a certain degree of proficiency in the language, as well as some non-trivial grammatical and morphosyntactical notions. For this reason, the term mhaggyānā has been interpreted also as designating the teacher of rhetoric (perhaps modelled upon the Greek meletáō "to meditate, to study"), while another hypothesis is that the activity of the mhaggyānā focused on vocabulary and on the difference between Syriac as a literary language and the dialects spoken by the pupils. [START_REF]A different approach to language sciences: orthoepy and punctuation 29 Sachau[END_REF] Two figures were in charge of more advanced teachings: the badoqā, "researcher" (from the root bdq "to search", but also "to show"), and the mpaššqānā, "interpreter" (from the root pšq "to explain, to interpret"). The first was specialised in secular knowledge, covering philosophy, history, geography, astronomy etc. The second, instead, was the exegete, who interpreted and commented on the Scriptures.

The Syriac world followed the subdivision of disciplines and the pedagogical cursus of the enkyklios paideia, a progressive learning pathway in which grammar, rhetoric (including poetry and metrics) and dialectic were meant to provide the essential analytical instruments to gain access to arithmetic, geometry, astronomy and music (the system designated in the West as trivium and quadrivium). 4 From this perspective, language sciences constituted a coherent system, sharing a technical vocabulary (a combination of native Aramaic terms, Greek loanwords and learned calques) and the same analytical method, since they all aimed at providing the pupil (as well as the proficient scholar) with the same linguistic key to a cosmos that was conceived as logically ordered/organised.

The Syriac Book of Dialogues, by the West-Syriac scholar Severos bar Šakko (d. 1241) testifies to the assimilation of this pedagogical and epistemological model. This encyclopaedic manual takes the form of questions and answers, and is arranged into two books: Book 1 consists of three dialogues on grammar, rhetoric and poetry respectively (in addition to a brief lexicographical appendix); Book 2 consists of two dialogues, one on logic and philosophy and the other on the definitions and subdivisions of philosophy, physics, mathematical sciences and metaphysics. Language sciences (grammar and rhetoric) are clearly considered as propaedeutic to all other disciplines.

A concurrent paradigm in the Syriac organisation of knowledge opposed the ecclesiastical sciences to the "external" or secular ones. Ecclesiastical sciences were the ones directly connected with the reading and interpretation of the Scriptures, theology, canon law etc., while all other disciplines were often designated as barāyā, meaning "external", but also "secular" (Mk 4,11). This subdivision partly mirrors the domains of expertise of the mpaššqānā and of the badoqā, the "external" sciences being the prerogative of the latter. 5 As far as language sciences are concerned, rhetoric and dialectic are always considered "external" disciplines, while the status of grammar seems to vary over time. Its position at the very beginning of Bar Šakko's Dialogues clearly places it among the "external" sciences; however, it can also belong to the ecclesiastical sciences, according to a perspective that considers grammar as a primary tool for accessing the biblical texts. By the end of Late Antiquity, the Aristotelian Organon was standardized by the Alexandrian tradition into a corpus comprising the canonical works Categories, Peri hermeneias, Prior and Posterior Analytics, Topics and Sophistical Refutations, together with Rhetoric and Poetics, the whole prefaced by Porphyry's Eisagoge. Between the sixth and the seventh century, an intense translation movement achieved the transfer into Syriac of this Greek logical corpus. Particularly, Categories, Peri hermeneias and Prior Analytics were the object of multiple West-Syriac translations (mostly by Proba, Jacob of Edessa, George bishop of the Arabs). In the same period, two translations were also produced of Porphyry's Eisagoge, the most notable one by Athanasius of Balad, who was also responsible for the first Syriac translation of Posterior Analytics, Topics and Sophistical Refutations. The translation activity was accompanied by the redaction of commentaries, such as those by Sergius of Rešʿayna (d. 536), Proba (sixth century), Paul the Persian (sixth century), Severus Sebokht (d. 666/7) and George Bishop of the Arabs (d. 724), which touched upon the first portion of the logical curriculum, from the Eisagoge to Prior Analytics. 6 In the same period, the adaptations of the Techne grammatike (sixth century, by Joseph Huzaya) [START_REF] Ed | Considerazioni interlinguistiche sull'adattamento siriaco della Techné Grammatiké di Dionisio Trace[END_REF] and of the Canons of Theodosius (seventh century, by Jacob of Edessa), [START_REF]Fragments of the Turras mamlla nahraya or Syriac Grammar of Jacob of Edessa[END_REF] provided Syriac students with the essential grammatical tools for accessing the Greek approach to language analysis. Such early works laid the conceptual and terminological foundations for the development of a rich grammatical tradition that was updated and adapted almost every century both by West and East Syriac authors, perpetuating the Greek model of metalinguistic analysis. [START_REF]For a history of the Syriac grammatical tradition see A. Merx[END_REF] With the development of Arabic linguistics, starting from the late eighth century, Syriac scholars living under Islamic rule were confronted with another paradigm of language analysis. During the early ʿAbbasid rule (ninth-tenth century) several Syro-Arabic lexicographical works were produced, following a movement of translation of Greek philosophical and scientific texts from Greek into Arabic, often via a Syriac intermediary. In the eleventh century the first Syriac grammar largely based upon an Arabic metalinguistic model was produced by the East-Syriac patriarch Elias of Ṭirhan (Elias I, d. 1049). During the twelfth and thirteenth centuries some Syriac grammarians rejected all Arabic influence (Yoḥannan bar Zoʿbi), while others attempted to integrate elements of Arabic linguistics within a traditional framework ultimately deriving from the Techne (Išoʿyahb bar Malkon, late twelfth-early thirteenth century; Barhebraeus, d. 1286). Among the main differences opposing the Greek-based Syriac grammars to the Arabic-based ones are: the number of the parts of speech (seven vs. three), the use of a metalanguage calqued upon Greek or Arabic terminology, and the prominence given in Arabic-based Syriac grammars to the morpho-syntactic issue of the distinction of agent and patient in the sentence.

The grammatical, logical and rhetorical corpus 2.1. Greco-Syriac translations and original compositions

Contrary to what we have seen for logic and grammar, no direct evidence has survived of early Syriac translations of the Greek rhetorical corpus. This fact could also be explained by the diffuse Greek-Syriac bilingualism of the early Syriac educational system, where translations were mainly meant for beginning students, whereas advanced content was probably studied in the Greek originals.

In the domain of sophistic rhetoric (concerning mainly the styles and structure of literary composition), [START_REF] Watt | Watt distinguishes between a sophistic and a logic branch of Greek rhetorical theory[END_REF] the earliest extant treatise is the Rhetoric by the West-Syriac Antony of Tagrit (ninth century?), [START_REF] Nicosia | Reassessing Antony of Tagrit: When Did he Actually Live?[END_REF] in five books, only partially edited, [START_REF]The Fifth Book of the Rhetoric of Antony of Tagrit[END_REF] followed by the second and the third of Bar Šakko's Dialogues. [START_REF]A brief outline of the contents in Bendrat, 'Der Dialog über die Rhetorik des Jakob Bar Shakko[END_REF] Antony's work appears to be a sort of school handbook. Book One, which is divided into thirty chapters and represents an introduction to philosophical rhetoric in its most technical aspects, provides a definition of rhetoric and of its components (1-3), and addresses praise and blame, persuasion and dissuasion, accusation and defence, representing the types of rhetorical discourses. [START_REF] Epideictic | Syriac Rhetorical Theory and the Syriac Tradition of Aristotle's Rhetoric[END_REF] The book discusses the composition of these discourses, the correct order to follow and their technical vocabulary (4-15). 15 Moreover, it deals with a series of rhetorical devices (16-21), types of prologues and epilogues (21), various utilities (22-24), the use of poetry, grammar and comedy in rhetoric (25-27), the sequence within a speech (28), [START_REF] Watt | Syriac Panegyric in Theory and Practice[END_REF] types of weaknesses in discourses (29), and it ends with an epilogue (30). [START_REF] Eskenasy | Complete chart[END_REF] Book Two discusses the benefits of praise; Book Three deals with subject matter, disposition and ornamentation, while Book Four concerns the praises of friendship; Book Five addresses poetry, namely metres, figures of speech and assonant letters.

Bar Šakko's Dialogues on rhetoric and poetry are largely based on the first and fifth books of Antony of Tagrit's Rhetoric, respectively. 18 The Dialogue on Rhetoric contains twenty-eight questions and answers, beginning with a definition of rhetoric (1) and the different species of rhetoric (2), a description of each species and of its internal articulation, e.g., panegyric, invective, exhortation etc. (3-8). The subsequent questions consider the techniques through which the art of rhetoric can be acquired (9, 11), further subdivisions of the different species (10, 12), and the sources from which exempla should be derived (14). Questions 16 to 26 address specific topics: the different sorts of prologues and epilogues ( 16), the advantages of following the different species of rhetoric (17), and the correlation between rhetoric and ecclesiastical law (18). 19 Questions 19-26 address occasional speech delivery: ecclesiastical feasts, consolations, epistles, narrative texts, and answers to questions. Questions 27-28 examine the necessary skills and the potential obstacles to mastering the art of rhetoric.

The subsequent Dialogue on Poetry 20 opens with a definition of poetry itself, with its four principal species: metres, assonant letters, narrative/plot and figures of speech (1-2). It addresses the elements composing the verse, its species (3-4), the species composing each type of syllabic metrical pattern (5), various types of verses (6-8), the laws governing assonant letters (9-10), and style suggestions (11). It discusses figures of speech, namely fable, aphorism, bare figure, prosopopoeia, metaphor/comparison (12-17), then narrative/plot (18), comedy (19) and tragedy (20), together with their species.

In the domain of philosophical rhetoric (concerning the different parts, classes and modes of discourse in connection with logic and dialectic), no extant manuscript hosts a Syriac translation Aristotle's Rhetoric, which led many to question its very existence. [START_REF]Above of all U. Vagelpohl, Aristotle's Rhetoric in the East: The Syriac and Arabic Translation and Commentary Tradition[END_REF] However, the author of the extant Arabic translation of the Rhetoric 22 declares that he used a Syriac version in the redaction of his version; this is apparent in his choice of Arabic technical words, which strongly points to the mediation of a Syriac translation. [START_REF] Nicosia | From Antony of Tagrit to the Arabic Version: the Syriac Technical Vocabulary of Rhetoric and the Migration of Words[END_REF] A section of Barhebraeus' Cream of Wisdom (1285-1286) contains the only extant commentary on Aristotle's Rhetoric in Syriac. 24 In seven chapters, the commentary considers 1. Usefulness of rhetoric and logical foundations of the discipline. 2. Deliberative rhetoric. 3. Epideictic rhetoric. 4. Judicial rhetoric. 5. Habits and passions of the soul 6. Common forms of the three species of rhetoric. 7. Rhetorical terminology and its organisation. The work is based on Ibn Sīnā's Arabic commentary hosted in the Kitāb al-Šifāʾ, [START_REF] Watt | al-mantiq VIII -al-ḫaṭāba (La logique VIII -La Rhétorique)[END_REF] and therefore it represents a synthesis of the Syriac and Arabic rhetorical traditions. Barhebraeus' commentary features a vocabulary which is clearly rooted in the Syriac tradition, and the lost Syriac version of Aristotle's Rhetoric seems to have been among its sources.

The two branches of Syriac rhetorical tradition, one represented by Antony of Tagrit and Bar Šakko and the other by Barhebraeus, feature an imposing number of Greek loanwords, and occasional Latin ones -mediated through Greek -which were integrated and adapted, and they often display new orthographies of previously known words. [START_REF] Nicosia | From Antony' and Nicosia, 'Tradition'. 27 See Hugonnard-Roche[END_REF]

The manuscript tradition

The manuscript tradition, especially East Syriac, displays a certain correlation between grammatical and logical texts. 27 The foundations of logic (Porphyry's Eisagoge, the first part of the Organon) are often accompanied by grammatical compendia. This is the case, for instance, with the manuscript Petermann 9 of Berlin Staatsbibliothek (1260), [START_REF] Sachau | Die Handschriftenverzeichnisse der königlichen Bibliothek zu Berlin[END_REF] which adds Syriac and Arabic translations and commentaries to texts such as Eisagoge, Peri hermeneias, Categories and Prior Analytics with various orthoepic texts (those focused on correct reading and reciting) and two different Syriac grammars, by Elias of Ṭirhan (based on the Arabic model) and by Bar Zoʿbi (based on the Greek model). The correlation is even stronger in first section of the manuscript, which displays a double-column layout, where Bar Zoʿbi's grammar and the Eisagoge are presented next to each other. Analogous cases, in much later manuscripts, are: Berlin Staatsbibliothek Sachau 226 (nineteenth century), 29 where various commentaries to Peri hermeneias and Prior Analytics are associated with the Syriac translation of the Techne grammatike; the manuscript 61 part B of the Chaldean Diocese of Alqosh (nineteenth century), [START_REF] Kessel | Field Notes on Syriac Manuscripts III. A Previously Unknown Philosophical Manuscript from Alqosh[END_REF] quite miscellaneous in content, but including the commentaries to Prior Analytics and Peri hermeneias by Severus Sebokht and by Proba, an anonymous Syriac translation of Prior Analytics, Proba's commentary on Peri hermeneias, together with grammatical texts by Dawid bar Pawlos (late eight -early ninth century) and some brief lexicographical texts. Such evidence defines a broader field of Syriac language sciences, crossing the boundaries between disciplines, in which grammar and logic constitute an organic and progressive system of learning, going from orthoepy and word-analysis, to morpho-syntax and discourse parsing, to dialectic and, ultimately, the classification of real-world phenomena, categories of thought and the relation between them. [START_REF]See also the manuscripts discussed in D. King 'Grammar and Logic in Syriac (and Arabic)[END_REF] Conversely, Antony of Tagrit's rhetorical work is usually transmitted on its own, [START_REF] Watt | XXV and Nicosia, 'Tradition[END_REF] with the exception of the most ancient extant manuscript hosting it, today split into two codicological units: British Library Add. 17208 + Deir al-Surian Syr. 32. [START_REF] Wright | Catalogue of the Syriac Manuscripts in the British Library[END_REF] This witness, dated to the ninth century, also features, together with the Rhetoric, a number of Antony's other compositions. These texts have not received a thorough study yet, but they have the aspect of exercises that Antony composed for himself or as examples for his students to follow. 34 As for Barhebraeus' Commentary to Rhetoric, the text was transmitted together with the other works composing his Cream of Wisdom, and therefore is always accompanied by the other commentaries to the Organon and, usually, to the rest of the Aristotelian corpus.

The most interesting cases are in the manuscripts of Bar Šakko's Dialogues: Oxford Marsh. 528 [START_REF] Smith | Catalogi codicum manuscriptorum bibliothecae Bodleianae. Codices Syriacos, Carshunicos, Mandaeos complectens[END_REF] and Jerusalem Saint Mark 233 36 lack the philosophical part of the Dialogues, but feature instead small treatises of grammatical content, such as discussions on points and verbal conjugations. Marsh. 528 also contains a few excerpts on religious matters and some folia from Elias of Nisibis' Nomenclator. Notably, manuscripts Marsh. 528 and British Library Add. 21454 37 host the only surviving fragment of the Syriac translation of Aristotle's Poetics. [START_REF] Margoliouth | Analecta Orientalia ad Poeticam Aristoteleam[END_REF] However, the other manuscripts preserve a variety of treatises and works together with the Dialogues, lacking any apparent overall connection. Moreover, the various parts of the Dialogues were often transmitted separately, according to their final use, for instance in schools, and to the needs of those who commissioned their manuscripts.

Besides the translation and redaction of grammars based on the Greek model, the Syriac linguistic tradition also developed a sophisticated orthoepic tradition, based on the use of a number of diacritical points that evolved over the centuries into an articulated system recording phonological, morpho-syntactical, prosodic and rhetorical information.

The defective Syriac writing system (an abjad, noting only consonants and some vowels through the non-systematic support of the letters noting the semi-vowels y, w and the consonants ʾ, h) gave rise to several homographs and ambiguous notations. Uncertainty in reading and the need to determine the correct interpretation of the Scriptures, gave rise to the earliest indigenous Syriac metalinguistic reflection, which took the form of textual annotation by means of diacritical marks (nuqzē, puḥḥāmē). Couples of homographs were distinguished by the use of a dot above or below the word. When a third homograph occurred, it was marked by a dot above plus a dot below. This simple and effective system evolved through the centuries with the notation of other phonological traits that could be crucial for the understanding of morpho-syntax: essentially vocalisation and lenition/redoubling of consonants. Punctuation was also used to convey prosodic and syntactical information. A simple system of three different signs (a sentence-final dot above the line, an intermediate dot separating two parts of one sentence, a lower dot dividing protasis from apodosis), was already in place in the fifth century. 39 These logical discourse marks developed into rhetorical accents which, through the combination of multiple dots, conveyed information about voice modulation and recitation. A list of thirteen early accents, already in use before the seventh century, has been drawn by Segal. 40 In addition to the main logical correlations, these accents expressed command, wonder, surprise, supplication, interrogation. In time, more sophisticated combinations of points were developed (Jacob of Edessa lists forty-seven accents). 41 All the positions and combinations of dots were given names, according either to their position or to their function. Such names constituted a sophisticated metalanguage (The net of points, according to the grammarian Išoʿyahb bar Malkon, twelfth century). It created a bridge between different language sciences and was described in treatises that, while teaching the basic notions of correct reading, condensed fundamental notions of grammar, logic and rhetoric.

3

  See the account in Barhadbšabba ʿArbaya, Cause de la fondation des écoles, ed. and trans. A. Scher, Patrologia Orientalis, T. IV, fasc. 4, n.° 18 (Turnhout, Brepols, 1981), pp. 349, 378-393 and the commentary by Scher on pp. 398-399. See also P. Bettiolo, 'Scuole e ambienti intellettuali nelle chiese di Siria', in C. D'Ancona (ed.), Storia della filosofia nell'Islam medievale. I. (Torino, Einaudi, 2005), pp. 48-10, and R. Contini, 'The role of linguistics in Syrian society', in S. Auroux et al. (eds.), History of the Language Sciences (Belin, De Gruyter, 2000), pp. 341-344. 4 Hadot, Arts libéraux, ch. VI (pp. 263-293); J. W. Watt, 'Grammar, Rhetoric and Enkyklios Paideia in Syriac', ZDMG 143 (1993), 45-71. On the role of rhetoric in propaedeutic education see J. W. Watt, 'A Portrait of John Bar Aphtonia Founder of the Monastery of Qenneshre', in J. W. Drijvers and J. W. Watt (eds.), Portraits of Spiritual Authority (1999), pp. 155-169. 5 V. Berti, Vita e studi di Timoteo I (m. 823) patriarca cristiano di Baghdad (Paris, Association pour l'avancement des études iraniennes, 2009), p. 81.

39

  Merx, Historia, pp. 64-65; J. B. Segal, The Diacritical Point and the Accents in Syriac, (London, Cumberlege, 1953), esp. pp. 58-65; E. J. Revell, 'Aristotle and the Accents. The Categories of Speech in Jewish and Other Authors', JSS 19/1 (1974), 19-35. 40 Segal, The Diacritical, pp. 64-65. 41 Jacob of Edessa, On Persons and Tenses, in A Letter by Mar Jacob, Bishop of Edessa, on Syriac Orthography; also a Tract by the Same Author, and a Discourse by Gregory Bar Hebraeus on Syriac Accents, ed. and trans. G. Phillips (London, Williams and Norgate, 1869), transl. pp. 13-33, ed. pp. *14-*24.

We are grateful to Riccardo Contini, Daniel King, Alberto Rigolio and Alexandra Lukes for their precious notes and suggestions.

See I. Hadot, Arts libéraux et philosophie dans la pensée antique (Paris, Études augustiniennes, 1984), ch. II.

H. Hugonnard-Roche, La logique d'Aristote du grec au syriaque (Paris, Vrin, 2004), pp. 11-16 ; H. Hugonnard-Roche, 'La tradition du Peri Hermeneias d'Aristote en syriaque, entre logique et grammaire', in M. Farina (ed.), Les auteurs syriaques et leur langue (Paris, Geuthner, 2018), pp. 55-93.