Christine Meunier
email: christine.meunier@lpl-aix.fr

Robert Espesser

Vowel reduction in conversational speech in French: the role of lexical factors

Keywords: Vowels, French, word frequency, word category, durations, spectral reduction, spontaneous speech

In this study we investigate vowel reduction and the role of some lexical factors in the production of vowels extracted from a corpus of French conversations. Vowel durations and spectral quality are examined with respect to 1. their interaction in the corpus, 2. the position of vowels in words, and 3. word frequency and word category. The analyses are conducted on vowels produced by 16 speakers. Our study provides strong evidence that vowel reduction (decrease in durations and more centralized spectral values) affects most of the vowels in conversational speech. The results show that vowels in final syllables of words were less often reduced while the preceding ones show reduced durations and centralized formant values. Moreover vowels are more reduced in monosyllabic function words than in monosyllabic content words. Nevertheless, we did not find a clear effect of word frequency on vowel durations. Finally, our study shows that vowel reduction depends on several factors related to lexical properties (word category) and to prosodic properties (stress and final lengthening).

Introduction

Speech reduction in conversational speech is currently a challenging topic, since its manifestation highlights the complex interaction and relationship between linguistic domains. It involves questions about the processes that lead speakers to produce speech in a non-canonical way. The main question is when (in the speech flow) and how (as which phonetic phenomena) speech is reduced. The answer to these questions would give us valuable information on the relationship between the phonetic, lexical, syntactic and pragmatic domains. This paper is focused on vowel reduction, which we consider to be a phenomenon of undershoot related to unattained targets, generally involving shortened vowel duration leading to a reduced vowel space. This phenomenon affects vowel quality so that the values observed under certain conditions are often far removed from prototypical ones. A large study on French vowel reduction has been conducted by [START_REF] Gendrot | Impact of duration on F1/F2 formant values of oral vowels: an automatic analysis of large broadcast news corpora in French and german[END_REF][START_REF] Gendrot | Impact of duration and vowel inventory size on formant values of oral vowels: an automated formant analysis from eight languages[END_REF]and 2008). These researchers investigated the reduction of French vowel duration and spectral quality on a large French corpus (ESTER) which is composed of recordings of French radio broadcast news, in which speech is largely prepared and planned. They showed that short vowels have more central spectral values, confirming that spectral reductions are correlated with duration reductions (as in other languages). Moreover, they highlighted the effect of some lexical factors on vowel durations. Our study is based on a corpus of conversational speech, since conversational speech is known to be less controlled than broadcast news [START_REF] Torreira | The Nijmegen Corpus of Casual Speech[END_REF]) and provides specific phenomena (disfluences, unfinished sentences, hesitations, back-channels, overlapping speech, etc.). Consequently, we ask whether vowel reduction would be greater or qualitatively different in very casual speech. Our study provides specific data on vowel production in casual speech in French.

The occurrence of reduction in vowel duration and changes in the spectral quality of the vowel is influenced by a number of factors. [START_REF] Lindblom | Spectrographic study of vowel reduction[END_REF] showed that vowel reduction occurs in languages with lexical stress, such as English and Swedish, and particularly affects unstressed vowels. It is generally agreed that French does not have lexical stress [START_REF] Hirst | Prosodic parameters of French: A cross-language approach[END_REF], and that stress in French is a property of the phrase rather than the word. Nevertheless, several studies have shown that the end of the Accentual Phrase (a domain greater than the lexical unit) coincides with a word-final syllable [START_REF] Dell | L'accentuation dans les phrases en français[END_REF][START_REF] Jun | Realizations of accentual phrase in French intonation[END_REF]. Since the end of an Accentual Phrase (AP) is the final syllable of a word, internal syllables should be more reduced than the final ones. Indeed, [START_REF] Adda-Decker | Contributions du traitement automatique de la parole à l'étude des voyelles orales du français[END_REF] showed that vowel duration depended on syllable position within the word: long vowels were more frequent in final syllables whereas short vowels tended to appear in non-final syllables. Since speech in our corpus is less controlled and prepared, we asked whether vowel durations and quality would be affected in the same way.

Speech communication is a dynamic process in which speakers adapt their production so that their message can be understood by listeners [START_REF] Lindblom | Explaining phonetic variation: a sketch of the hyper-and hypospeech theory[END_REF]). Reduced speech therefore often appears when the linguistic message is highly predictable [START_REF] Jurafsky | Probabilistic relations between words: Evidence from reduction in lexical production. Frequency and the emergence of linguistic structure[END_REF][START_REF] Hume | Deconstructing markedness: A Predictability-based Approach[END_REF]). On the one hand, the predictability of words may be determined in part by their frequency of occurrence in speech. Several studies have shown that high frequency words are more often reduced than low frequency ones [START_REF] Bybee | Word frequency and context of use in the lexical diffusion of phonetically conditioned sound change[END_REF]. [START_REF] Pluymakers | Lexical frequency and acoustic reduction in spoken Dutch[END_REF], for example, observed reduced durations in high frequency morphologically complex Dutch words. But word frequency also affects phonetic details such as voicing assimilation [START_REF] Ernestus | Lexical frequency and voice assimilation[END_REF]. On the other hand, predictability may also depend on word category. In a corpus of spontaneous conversations between American English speakers, [START_REF] Johnson | Massive reduction in conversational American English[END_REF] observed that function words were deleted (or produced with forms that deviated from their citation form) more often than were content words. In a study on controlled speech in Dutch, van [START_REF] Van Bergem | Acoustic vowel reduction as a function of sentence accent, word stress, and word class[END_REF] showed that both word stress and word category have an influence on the spectral quality of vowels: unstressed words and function words were more reduced. From this overview, it seems that the production of reduced forms, and particularly acoustic reduction, is strongly related to predictability and information in the speech flow. The more predictable a word is, the less phonetic information is necessary [START_REF] Hume | Deconstructing markedness: A Predictability-based Approach[END_REF]. With respect to vowel reduction, [START_REF] Adda-Decker | Contributions du traitement automatique de la parole à l'étude des voyelles orales du français[END_REF] showed that short vowels are more often present in function words than in content words. In our analysis we quantify this difference and we analyze spectral quality according to both word categories (function word and content word).

Speech reduction and variation has received particular interest since studies on spontaneous speech have become more common. It has been shown that the amount of reduction in spontaneous speech is greater than expected [START_REF] Johnson | Massive reduction in conversational American English[END_REF]. Speaking styles differ in what reduction phenomena they demonstrate, depending on the degree of control in the speech. In our study, we aim to evaluate the amount of vowel reduction in very casual speech. We start with a general overview of vowel reduction in our corpus, and then present our analyses of the lexical factors.

Speech material and analyses

The Corpus of Interactional Data (CID)

The CID corpus [START_REF] Bertrand | Le CID. Corpus of Interactional Data. Annotation et Exploitation Multimodale de Parole Conversationnelle[END_REF]) consists of eight hours of audio/video recordings of French conversational speech. Each hour is a recording of a conversation between two participants, speakers from the southeast region of France or long-term residents of the region. Five conversations are between two women and three are between two men. The conversations took place in a studio, and the participants wore head-mounted microphones, with each voice recorded on a separate track. The participants were given two topics to discuss: conflicts in their professional environment or funny situations that they had found themselves in. These were only suggestions however, and the participants were free to talk about any topic. As the speakers knew each other quite well, the resulting recordings are highly relaxed conversations.

The corpus was transcribed orthographically by two advanced phonetics students. This transcription was augmented with certain phonetic details (for example, particular pronunciations, such as je sais 'I know', canonically pronounced /Ћəsϯ/, but often pronounced [Ѐϯ], were included in the transcription, as well as regional pronunciations). This initial transcription was then processed by a grapheme-to-phoneme converter (Di Cristo & Di Cristo 2001) and an aligner [START_REF] Fohr | The automatic news transcription system: ANTS, some real time experiments[END_REF]) to get the time point of the segments with the speech signal.

The CID corpus is partially or fully annotated on several linguistic levels (segmental, prosodic, syntactic, discourse, gestural, etc.) allowing the analysis of the phonetic segments in interaction with the other linguistic levels. The results presented in this paper are based on the automatic alignment for all 16 speakers.

French vowels in the CID

The French vowel system includes 11 oral vowels and three or four nasal vowels, depending on the dialect. We are interested in global characteristics of vowel reduction, and our hypotheses do not at all depend on the distinction between oral and nasal vowels. The spectral analysis of nasal vowels is often not reliable, and we therefore chose to analyze only the oral vowels. French does not have a phonological distinction between long and short vowels. Oral vowels represent 39% of all the phonemes in the CID corpus (106,961 oral vowels out of 272,166 phonemes). Although 11 oral vowels are usually distinguished in French, the aligner recognizes only seven macro-classes of oral vowels, since it does not distinguish between mid-closed and mid-open vowels (see Table 1). We do not consider this to be problematic for our analysis, since these distinctions are not strictly phonological. The distinctions between mid-closed and mid-open vowels (/e/-/ϯ/, /ø/-/oe/, /o/-/ѐ/) could be considered as allophonic variation, since these vowels do not typically appear in the same syllable structures. The open vowels appear in closed syllables (nerf 'nerve' /nϯr/), while the closed vowels appear in open syllables (nez /ne/ 'nose'). For some cases, minimal pairs (épée /epe/ 'sword', épais /epϯ/ 'thick') can be found, but not in all regional accents [START_REF] Durand | Structure et variation dans quelques systèmes vocaliques du français: l'enquête Phonologie et Français Contemporain (PFC)[END_REF]. The consequences for the analyses are that /e/, /o/ and /ø/ would show intermediate spectral values in our corpus. The vowel /e/ is the most frequent vowel in our corpus followed by /a/, /ø/ and /i/. Together /e/, /a/ and /ø/ represent 71% of all the oral vowels. The least frequent oral vowels are /o/, /y/, and /u/. This distribution is quite similar to that described in literature on French [START_REF] New | Une base de données lexicales du français contemporain sur internet: LEXIQUE[END_REF], although in other studies, /a/ is typically the most frequent vowel. These differences between the vowel frequencies reported in the literature, and what is observed in the CID corpus can be explained by the merging of the mid-vowels in the automatic alignment. Most of the vowels are produced in a very few words (frequent monosyllabic function words), with 50% of the vowels realized in fewer than 70 different words. For example, 11% of the tokens of /u/ in the corpus appear in the function word tout 'every, all'.

Analyses

Vowel durations were computed from the boundaries placed by the aligner. Due to the parameterization of the aligner, the minimal duration was 24 ms and changed by a step of 8 ms. Formant frequencies were estimated by a linear prediction method (autocorrelation), with a Viterbi algorithm to select the best candidates by imposing frequency continuity constraints (ESPS package, Entropic, 1997). A total of 32,266 vowels shorter than 30 ms and 2,075 vowels longer than 300 ms were dropped from the analyses. These very short vowels often correspond to alignment issues caused by transcriptions in which phonemes not physically present in the signal were transcribed. Extremely long vowels often correspond to long hesitations in the production of certain function words (mais 'but', alors 'so', etc.). The exclusion of these very long vowels did not affect the analysis, though, since they were not within the scope of the study. As a result, in this first section, the analyses will be conducted on the 72,620 remaining oral vowels.

All the analyses were carried on using linear mixed effects models (LMEs) which provide a powerful tool for the analysis of grouped data. A mixed model incorporates both fixed effects, which are parameters associated with certain repeatable levels of experimental factors, and random effects, which are associated with individual experimental units sampled at random from a population [START_REF] Pinheiro | Mixed-Effects Models in S and S-PLUS[END_REF][START_REF] Quené | On multi-level modelling of data from repeated measures designs: A tutorial[END_REF]. Hence LMEs take into account the correlation of the observations within the same experimental unit. Moreover LMEs can handle unbalanced data. The question of degrees of freedom for these models continues to be a subject of some controversy in the literature. Hence a Monte Carlo Markov Chain technique [START_REF] Baayen | LanguageR: Data sets and functions with Analyzing Linguistic Data: A practical introduction to statistics[END_REF][START_REF] Bates | lme4: Linear mixed-effects models using S4 classes[END_REF] is used to get p-values (MCMC pvalues). The data analysis was carried on using the computing language R (R Development core Team, 2008)

Results

Durational and spectral values

The vowel durations extracted from our spontaneous speech corpus are extremely short compared with the durations of vowels produced in controlled speech (e.g., in the reading of nonsense CVC and VCV sequences by French speakers, [START_REF] Bartkova | A model of segmental duration for speech synthesis in French[END_REF], Table 2). The mean vowel duration was 73 ms.

The longest vowel was /u/, followed by /ø/ and /i/. The shortest vowel was /a/ (Table 2). The distribution of vowel durations was highly unbalanced: 50% of vowel durations were less than 60 ms.

A high degree of reduction in vowel durations was thus observed in this corpus, in line with the results observed in another study on French [START_REF] Adda-Decker | Contributions du traitement automatique de la parole à l'étude des voyelles orales du français[END_REF].

We next investigated the relation between reduction in duration and spectral reduction. The table 3 shows the spectral values of the oral vowels of the CID. Male and female formant values are presented separately, along with a reference [START_REF] Tubach | La parole et son traitement automatique[END_REF]. We present the mode (the higher mode, if there were two modes) of each formant value distribution, since the large amount of data allows the use of this robust measure. The number of measures across the different cases ranged from 196 (long /u/ male speakers) to 6655 (short /e/ female speakers). The vowels were split into three different duration-dependent data sets corresponding to short, intermediate and long vowels in the CID corpus: Set 1=30 ms to 50 ms, Set 2=51 ms to 120 ms, Set 3=121 ms to 300 ms.

Table 3: Female and male spectral values (F1, F2 and F3 in Hz) according to the three duration sets in the CID. Values in controlled speech (Tubach, 1989) are given for a comparison. The values in parentheses indicate that the second higher mode is used (and was selected after visual inspection of the distribution). The number of measurements in each set (female and male values

) is also given. 1). F1 values of mid-vowels also decreased to a lesser extent. The F1 of closed vowels was not sens duration. We thus observed a decrease of F1 values related to a duration decrease for mid open vowels, particularly for the open vowel /a/. The variations of F1 show a trend for speakers to produce short vowels with a less open jaw and/or higher front vowels (/i/ and /e/), an increase especially for back vowels (/u/ and /o/) and, to a lesser extent, for central vowels (/a/ and /ø/). Thus the production of short vowels appears to be le back vowels show more central values). female vowel space and tends to have greater reduction for back vowels than for front ones (figure Note that, for both male and female speakers, the first three formants of the four vowels /a/, /e/, / are extremely close for short durations (set 1, Table 3). This lack of acoustic separation may also suggest that these four vowels may be difficult for listeners to dist front vowels (/i/ and /e/) as duration decreases, while the other vowels show an increase of F3 for short vowels. F3 variations tend to show that, for short vowels, lip position may be less extreme (less rounde for rounded vowels and less spread for unrounded vowels). . Thus the production of short vowels appears to be less peripheral (front and back vowels show more central values). The results show that the male vowel space is more reduced than female vowel space and tends to have greater reduction for back vowels than for front ones (figure and female speakers, the first three formants of the four vowels /a/, /e/, / are extremely close for short durations (set 1, Table 3). This lack of acoustic separation may also suggest that these four vowels may be difficult for listeners to distinguish. A decrease in F3 is observed for the front vowels (/i/ and /e/) as duration decreases, while the other vowels show an increase of F3 for short vowels. F3 variations tend to show that, for short vowels, lip position may be less extreme (less rounde for rounded vowels and less spread for unrounded vowels). ss peripheral (front and The results show that the male vowel space is more reduced than female vowel space and tends to have greater reduction for back vowels than for front ones (figure 1).

and female speakers, the first three formants of the four vowels /a/, /e/, /ø/ and /o/ are extremely close for short durations (set 1, Table 3). This lack of acoustic separation may also suggest A decrease in F3 is observed for the front vowels (/i/ and /e/) as duration decreases, while the other vowels show an increase of F3 for short vowels. F3 variations tend to show that, for short vowels, lip position may be less extreme (less rounded of F1 and F2 values in 120ms, Set 3: 121-300ms).

These results are in agreement with those obtained in previous studies [START_REF] Lindblom | Spectrographic study of vowel reduction[END_REF][START_REF] Gendrot | Impact of duration on F1/F2 formant values of oral vowels: an automatic analysis of large broadcast news corpora in French and german[END_REF] and confirm the relationship between decrease in vowel durations and hypoarticulation. Although the comparison is not perfect (in Gendrot & Adda-Decker, 2005, only the mean formant values are given), it seems that the spectral reduction observed is greater in our corpus, especially for /a/ which is more reduced in the CID than in Gendrot & Adda-Decker's study. Moreover, our results provide different patterns of reduction in male and female speakers' productions.

Lexical factors

In this section, we discuss duration analyses conducted on all vowel data, since duration variation affects all vowels in the same way. Spectral analyses, however, were not conducted on all vowels for several reasons. First, vowel spectral data have to be analyzed separately for each vowel (the variation observed depend on spectral parameters) and this complicates the analyses and their interpretations. Second, spectral variation depends on vowel identity, and certain analyses may be irrelevant for particular classes of vowels (for closed vowels like /y/ and F2 for the central vowel /ø/, for example). Third, in this section, we will focus not on vowel-specific effects, but rather on lexical factors. We therefore carried out spectral analyses on the F1 of /a/, since we observed particularly striking spectral variation related to duration variation for this vowel (Figure 1). Furthermore, there are many tokens of /a/ since it is the second-most frequent vowel in our corpus. Therefore we considered the analysis of F1 of /a/ to be a good starting point to provide results of vowel spectral reduction according to lexical factors.

Within-word position effects

Several studies [START_REF] Lindblom | Spectrographic study of vowel reduction[END_REF][START_REF] Van Bergem | Acoustic vowel reduction as a function of sentence accent, word stress, and word class[END_REF] have shown that vowels in unaccented syllables are more often reduced than those in accented syllables. This seems to be a particularity of languages with lexical stress. French does not have lexical prosodic characteristics [START_REF] Hirst | Prosodic parameters of French: A cross-language approach[END_REF], and there is no lexical stress in the language. The location of stress is fixed at the word level, but the domain (the Accentual Phrase, AP) of its realization can be larger than the word (Jun & Fougeron, 2002 and references therein). Consequently, the final strong syllable (not including word-final schwa) of a word is realized with a longer duration only if it is the end of an AP [START_REF] Dell | L'accentuation dans les phrases en français[END_REF]. The AP may thus correspond to single word or to a group of words. Consequently, our hypothesis is that reduced vowels will appear in non-final syllables of words. Since we do not have a complete annotation of the APs in the CID corpus, vowels were analyzed according to their syllable position in the word. Each word was syllabified individually on the basis of its actual phonetic realization (for example, if the bisyllabic word petit 'small' /pђ.ti/ was realized with the vowel /ђ/ elided, the vowel /i/ was therefore embedded in the monosyllabic item /pti/). We considered five different sets of words, defined by word length (1, 2, 3, 4 and 5 syllables), and we examined the target vowels at each syllable position within the word. For example, /i/ is in second syllable in the three-syllable word papillon 'butterfly', and /y/ is in first syllable in the monosyllabic word tu 'you'. The number of vowel tokens examined for each set is given in Table 4. Words with 6 or 7 syllables were disregarded, given their very low frequency in the corpus. For each set, we fitted a linear-mixed model (LME), with the logarithm of the vowel duration as the dependent variable. The fixed effect was the position of the vowel in the word, expressed as an n-level factor for a n-syllable word. Random intercepts were included for speakers.

Table 4 Number of vowels according to word length (1, 2, 3, 4 and 5 syllables).

Number

Figure 2 shows estimated vowel durations in the different conditions. The results show that vowels of the final and penultimate syllables of words are longer than those of earlier syllables. As an example, for 3-syllable words, the vowel of the second syllable (54 ms) was significantly longer than the vowel in first position (49 ms); similarly, the vowel in third position (68 ms) was significantly longer than the vowel in second position (54 ms). The significant p-values (MCMC p-values) were all less than 0.003, while the non-significant p-values were all greater than 0.46.

The observed duration increases were quite regular. The mean durations of the final syllable vowels were very similar (64-68 ms), regardless of the number of syllables in the word. The one exception was the set of 5-syllable words, for which the final syllable was longer (77 ms). The mean duration of vowels in monosyllabic words was similar to that of vowels in the final syllables of multisyllabic words. In our study, the estimated duration of the final syllable is the mean duration across unstressed and stressed final syllables. Consequently, as the unstressed syllables are not lengthened, at least relative to the AP, it is important to interpret these lengthening values with caution. The real values of stressed vowel lengthening are thus probably longer than the values we mentioned. Our results are consistent with a recent description of French radio speech; [START_REF] Adda-Decker | Contributions du traitement automatique de la parole à l'étude des voyelles orales du français[END_REF] observed that long syllables (>100ms) are more often in final syllables while short ones (<50ms) are more often in non-final ones. Moreover, we observed that vowel lengthening also affects the penultimate syllable as well. An explanation could be that the stress for word with final schwa would fall on the penultimate syllable (since schwa cannot be stressed). But a complementary study (where words with final schwa were excluded) shows the same trends. In fact, the lengthening of the penultimate syllable has been observedbut not quantified -in a previous study [START_REF] Di Cristo | De la Microprosodie à l'Intonosyntaxe[END_REF]. This may reflect a tendency for this lengthening phenomenon to cross syllable boundaries.

In addition to Adda-Decker's study, we analyzed formant values in order to observe if vowel quality is affected by duration variation within words, and thus by their position in the words. As noted above, we focused on spectral variations of the F1 of /a/. We eliminated the 5-syllable word set, since there were too few tokens (0.5% of the total) containing /a/ in the final syllable. A total of 16,664 /a/ tokens were analyzed (Table 5). For each set, we fitted a linear-mixed model, with the logarithm of the F1 (in Hz) value as the dependent variable. The fixed effect was vowel position in the word expressed as an n-level factor for an n-syllable word. Random intercept terms were included for speakers.

The results are quite consistent with the lexical pattern observed for durations. F1 values are significantly higher for /a/ in final syllables (Figure 3), consistent with lower tongue position and/or a more open jaw, while F1 for /a/ in non-final syllables was lower, and thus farther from prototypical values for this open vowel (Table 1,F1). For each set, the F1 of the last vowel differs significantly from the preceding one. The significant p-values were all less than 0.001, while the non-significant p-values were all greater than 0.067. The vowel of the penultimate syllable was reduced like the earlier ones. The very small, although significant, duration differences observed between penultimate vowels and preceding ones were probably not sufficient for formant differences to emerge for this specific vowel (/a/).

We note a difference between reduction in duration and spectral reduction: F1 values for /a/ in monosyllabic words were similar to those for /a/ in non-final syllables, while durations of vowels in monosyllabic words were similar to those of vowels in final syllables. Consequently, vowels in monosyllabic words seem to have reduced spectral values but unreduced durations. The final lengthening can be interpreted as an effect of the final stress in the AP. This stress is marked on the final syllable (of the word), therefore the final syllable is more likely to be stressed and thus lengthened. Our results do not demonstrate that final vowels are never reduced since some of them might be reduced (within the AP domain) and some others are not (end of the AP). Nevertheless, our data shows that vowels in non-final syllables are reduced. This reduction affects the durations of vowels (as shown by Adda-Decker in a different corpus) but also vowel spectral quality, as the decrease of the F1 of /a/ in our study suggests.

Effect of word frequency and word category

The type of word used in conversation may also have an influence on how vowels are realized, according to the frequency of words and their informativeness. Frequency and informativeness are lexical factors which may influence phonetic reduction: when a word is highly frequent and also highly predictable, listeners may need less phonetic information to identify it. Similarly, words which are poorly informative (conjunctions, prepositions, etc.) and syntactically highly predictable in an utterance may be reduced without disturbing the listener's comprehension of the message. Indeed, Adda-Decker (2008) found that vowels in function words are more often short than vowels in content words.

To investigate this question, we analyzed vowel reduction according to two lexical factors: word frequency and word category. We computed word frequencies based on the CID corpus itself since frequencies computed from conversational speech might differ from those of written texts or more controlled speech (newspapers or broadcast news for example). Word frequency was defined as: the number of token occurrences/the total number of tokens in the corpus (about 110,000 tokens). We analyzed only monosyllabic words in order to control the variation in duration due to the position of the vowel within the word (see section 3.2.1."Within-word position effects"). Oral vowels in monosyllabic words represent 57% (41,592 vowels) of the total number of oral vowels in the corpus (72,620).

We distinguished two word categories: content words (nouns, verbs, adjectives and adverbs) and function words (prepositions, conjunctions, pronouns and auxiliary verbs). Content words have a higher semantic load than function ones. These two categories represent 84% of the monosyllabic productions. Other productions (interjections like euh, ah, ben, hum, etc.) were not taken into account. There were 25 monosyllabic structures in the CID corpus, however the three most frequent structures accounted for 81% of the monosyllabic words: CV (45%, n=15,708), V (24%, n=8,275) and CVC (12%, n=4,414). Moreover, the V structure was excluded from the analyses. The phonetic neighborhood of a vowel in a V syllabic structure is obviously unpredictable. We reasoned that a variable phonetic context and a possible resyllabification by speakers might cause specific variations of duration which are not relevant for our analysis. For instance, the sequence il est parti 'he has gone', produced /i.le.par.ti/ with /e/ in a CV syllable, but considered as a V syllable, since tokens are syllabified independently. Rare phonetic sequences, i.e. less than five occurrences in the corpus (about 1.8% of the vowels), were excluded. A total of 19,312 vowels distributed over 202 phonetic sequences remained (Table 6). With this type of corpus, word occurrences cannot be controlled and are therefore highly unbalanced. For instance, the 4th quantile of the lexical frequency distribution of the CVC function words contained 380 tokens. But these 380 tokens were all occurrences of a single word type, pour /pur/ 'for'. Similarly, considering the distribution of the lexical frequency of the CV function words, 95% of the tokens with a lexical frequency above the median is represented by only 6 word types. Models with lexical frequency taken as a continuous parameter could therefore be biased.

We therefore transformed lexical frequency to a 2-level factor (henceforth dichotomized lexical frequency or DLF). Within each of the four subsets of Table 6, this factor is equal to H (high) if the lexical frequency is above the median value of the subset, L (low) otherwise. We ran separate analyses for CV and CVC words, since there were too few different CVC function words (types) with a high lexical (token) frequency to safely analyze both the effects of lexical frequency and word category for the CVC set.

CV words

We ran a LME with the logarithm of the vowel duration as the dependent variable, the dichotomized lexical frequency (DLF) and the word category as fixed effects. Random intercepts were added to account for the variability across the 16 speakers and the 75 phonetic sequences. The interaction coefficient between the DLF and the word category was not significant (MCMC p-value= 0.16). A model without this interaction term showed that DLF has no effect (MCMC p-value = 0.9), and that the morphosyntactic category is highly significant (β=-0.19; se=0.018; t=-10; MCMC p-value <0.001). The estimated vowel duration was 57 ms for the function words and 69 ms for content words (Figure 4).

CVC words

As mentioned above, the influence of lexical frequency on duration can only be analyzed for the content words. We first studied the influence of the lexical frequency on the content words, and then analyzed the word category effect.

We ran a LME with the logarithm of vowel duration as the dependent variable and the DLF as the single fixed effect. The random terms were the same as those in the CV model (there were 117 phonetic sequences). The DLF was significant (β=0.11; se=0.037; t=2.98; MCMC p-values < 0.01). The estimated vowel duration was 64 ms for the high frequency words, and 71 ms for the low frequency words. This difference is slightly below the value of the resolution of the aligner (8 ms), and therefore must be interpreted with caution.

We investigated the influence of the word category separately, with a LME with the single factor word category as fixed effect. The random terms were the same as those in the CV model (there were 124 phonetic sequences). The word category factor was clearly significant (β= -0.208; se=0.026: t=-7.9; MCMC p-value < 0.001). The estimated vowel duration was 56 ms for function words, and 71 ms for content words (Figure 4). These values are very close to those obtained for the CV words.

We then asked whether these duration differences would correspond to spectral differences. We then examined variations of the F1 of /a/. The influence of word frequency on the F1 of /a/ was not analyzed, since we did not observe an effect of lexical frequency on vowel duration. We therefore focused on the influence of word category on F1. There were 5,360 monosyllabic words with the /a/ vowel (2,545 function words, 2,815 content words).

An LME was estimated, with the logarithm of F1 as the dependent variable and word category as fixed effect. Random intercepts were added to account for the variability across the 16 speakers and the 31 syllabic structures. The effect of word category was significant (β= -0.111; se=0.0158; t=-6.96; MCMC p-values <0.001). The estimated F1 values appear in Figure 4.

Figure 4: Estimated vowel durations (left in ms) and estimated F1 of /a/ (right in Hz) in monosyllabic words according to Word Category

To summarize, we did not find a clear effect of word frequency on vowel durations. A few words were very frequent in the corpus, and this complicated the analysis. But frequency and word category are highly related: most function words are very frequent, while most content words are rare. On the other hand, a clear effect of word category was found: content words show longer vowel durations and higher F1 of /a/, while function words show shorter vowel durations and lower F1 of /a/. Thus, as expected, function words are more reduced than content words. Nevertheless, once again several factors may interact: function words appear less often at the end of the AP in French (although they may occur in this position in spontaneous speech, as in t'es où? 'where are you?', tu viens là? 'are you coming here?'). Consequently, the reduction of function words may be attributed to their specific category but also to their position in the stress domain.

Discussion

Our goals were 1. to provide an overview of vowel reduction in casual speech in French and 2. to evaluate a possible effect of lexical factors on vowel reduction. The factors examined were the position of the vowel within the word, on one hand, and lexical frequency and word category, on the other.

Firstly, our study, based on conversational speech, confirms a recent study comparing French with other languages [START_REF] Gendrot | Impact of duration and vowel inventory size on formant values of oral vowels: an automated formant analysis from eight languages[END_REF], and showing that vowel duration and spectral quality are highly related. Moreover, our study provides a precise quantification of vowel reduction and shows that spectral reduction is more pronounced in casual speech.

Secondly, the study provides clear evidence of a relationship between vowel position in the word on one hand, and duration and spectral reduction, on the other. Due to the Accentual Phrase, vowels in the final (and, to a lesser extent, penultimate) syllables of words may be lengthened, while earlier vowels have shorter durations. In another study on French speech, Adda-Decker et al. ( 2008) found similar results, except for penultimate syllables. Moreover, our results show that the spectral quality of vowels is also affected by the position of the vowel in the word: F1 values of /a/ are lower in non-final vowels, suggesting that non-final vowels are reduced in multisyllabic words. Previous studies show that unstressed vowels are reduced in languages with lexical stress like Swedish [START_REF] Lindblom | Spectrographic study of vowel reduction[END_REF] or Dutch [START_REF] Van Bergem | Acoustic vowel reduction as a function of sentence accent, word stress, and word class[END_REF]. We find the same results in French, although the language does not have lexical stress.

Thirdly, our results show that the production of French vowels depends on word category. Vowels produced in monosyllabic function words were more reduced than those produced in monosyllabic content words. These results are in agreement with those of van Bergem (1993) for Dutch vowels observed in controlled speech. As van Bergem notes, it is difficult to determine whether the production of reduced vowels in function words is due to the small "semantic load" of these words or to their high frequency of occurrence. He concluded that it might be due rather to high frequency because a frequency effect is also observed across other word categories in his study. Our results for French, however, are not in agreement with these conclusions, or with other previous results showing frequency effects (Pluymaekers et al., 2005;[START_REF] Ernestus | Lexical frequency and voice assimilation[END_REF]. According to these studies, we expected that frequent words would be produced with reduced vowels in French, but our findings did not confirm an effect of frequency on vowel reduction. When words were relatively balanced across the different categories, no clear frequency effect was observed.

The analyses may be biased by the very high frequency of some function words in the CV and CVC structures, since spontaneous speech corpora are obviously not controlled. Our study concentrated on monosyllabic words, which are very frequent in the corpus (57% of the words of the corpus are monosyllabic, and the most frequent words are monosyllabic). It was impossible to analyze bisyllabic or longer words to compare function words and content words, since the number of function words is very low in these structures. There are only a very few monosyllabic types of function words, but these types are very frequent. At the other extreme, the number of types of content words is very high, but most of these types occur infrequently. How therefore should duration variations be interpreted? These variations might be due to the durations of a specific vowel if this vowel is present in a very frequent word, or to specific phonetic sequences (phonetic context) in monosyllabic words (for example, 75% of the /a/ tokens in monosyllabic function words appear in only two words). How can a very few words with many repetitions (tokens) be fairly compared to a great number of words with very few repetitions? In the same way, the effect of word category (vowels in function words are more reduced than those in content word) is hard to attribute to the single category factor, since function words appear more often in unstressed position. These are the limits of some analyses on natural speech. Some data are difficult to compare because they are too 'unbalanced' in natural speech. Studies on non-controlled speech (i.e. conversational or spontaneous speech) are obviously constrained by the real distribution of words in speech.

To summarize, we showed a significant effect of syllable position and word category on vowel reduction. Nevertheless, we must be careful not to over-interpret these effects, since the size of the variations observed seems to be only a small part of the general variation in the vowels of the corpus. Spontaneous speech is characterized by specific pragmatic and discourse functions. We hypothesize that a great part of vowel reduction may be conditioned by prosody and discourse constraints [START_REF] Beyssade | Prosody and Information in French[END_REF]. We are particularly interested in investigating in more depth the prosodic and spectral characteristics of the vowels of function words. Some function words, such as mais 'but' or et 'and' have particular discourse functions (hesitation, emphatization, etc.), and we hypothesize that vowel reduction in spontaneous speech may be even more complex than a simple distinction between function words and content words.

Figure 1 :Fomant

 1 Figure 1: Vowel triangles for female (l Hz) according to durati

  Vowel triangles for female (left) and male (right) speakers (mode of F1 and F2 values Hz) according to duration sets (Set 1: 30-50ms, Set 2: 51-120ms, Set 3short vowels) the vowel space is clearly less peripheral than in Set 3 (long vowels), with Set 2 being intermediate. The extent of the tral parameters (F1, F2 or F3). Concerning F1, the most sensitive spectral reduction was the decrease of /a/ values, the most open vowel (Figure 1). F1 values of itive to reduction in . We thus observed a decrease of F1 values related to a duration decrease for mid open vowels, particularly for the open vowel /a/. The variations of F1 show a trend for speakers to produce short (less low) tongue position. F2 values show a clear decrease for front vowels (/i/ and /e/), an increase especially for back vowels (/u/ and /o/) and, to a lesser extent, for

Figure 2 :

 2 Figure 2: Vowel durations (ms) estimated by the model. Durations are plotted as a function of position of the vowel in the word and of word length. The four curves represent vowel duration according to the position of the vowel in words from 2 to 5 syllables long. The isolated unfilled circle represents monosyllabic words. Filled circles indicate that the value differs significantly from the preceding one.

Figure 3 :

 3 Figure 3: Estimated F1 of /a/ as a function of its position in the word and of vowel length. The three curves represent F1 according to the position of the vowel in words from 2 to 4 syllables long (F1 values for /a/ in monosyllabic words and in the first position of bisyllabic words are identical). Filled circles indicate that the value differs significantly from the preceding one.

Table 1 French oral vowels in IPA and vowels grouped by the aligner in the CID

 1

	IPA	a	e	ϯ	o	ѐ	ø	oe	ə	i	y	u
	Aligner	a	e		o			ø		i	y	u
	Number of vowels in the CID	25057 (23%)	34915 (33%)	7178 (7%)	15703 (15%)	13337 (12%)	6010 (6%)	4761 (4%)

Table 2 : Vowel mean durations (ms) in the CID corpus in comparison with vowel mean durations in controlled speech (Bartkova & Sorin 1987)

 2

		a	e	ϯ	o	ѐ	ø	oe	ə	i	y	u
	CID	65	71		73			84		75	72	86
	Bartkova & Sorin (1987)	177	180	175	186	170	186	185	130	170	167	170

Table 5 Number of /a/ tokens according to word length.

 5

			Number of syllables in word
		1	2	3	4
	Number of /a/ tokens	9,830 (59%)	4,809 (29%)	1,401 (8%)	624 (3.5%)

Table 6 number of vowels in the two word categories according syllabic structure.

 6

		Content words	Function words	Total
	CV	3,276	12,538	15,814
	CVC	2,256	1,242	3,498


Mathieu Lefort

Alexander Gepperth

Learning of local predictable representations in partially learnable environments

PROPRE is a generic and cortically inspired framework that provides online input/output relationship learning. The input data flow is projected on a self-organizing map that provides an internal representation of the current stimulus. From this representation, the system predicts the value of the output target. A predictability measure, based on the monitoring of the prediction quality, modulates the projection learning so that to favor learning of representations that are helpful to predict the output. In this article, we study PROPRE when the input/output relationship is only defined in a small subspace of the input space, that we define as a partially learnable environment. This problem, which is not typical of the machine learning field, is however crucial for the robotic developmental field. Indeed, robots face high dimensional sensory-motor environments where large areas of these sensory-motor spaces are not learnable since a motor action cannot have a consequence on every perception each time. We show that the use of the predictability measure in PROPRE leads to an autonomous gathering of local representations where the input data are related to the output value, thus providing good classification performance as the system will learn the input/output function only where it is defined.

I. INTRODUCTION

According to sensory-motor theories, sensory-motor regularities that emerge from the interaction between an agent and its surrounding environment are a key point for this agent to build perceptions of the environment [START_REF] Mossio | Action-dependent perceptual invariants: From ecological to sensorimotor approaches[END_REF]. Moreover, they seems to play a crucial role for humans as various experiments in psychophysics show that stimuli consistent with usual environmental interactions improve learning and detection of events compared to unisensory or inconsistent stimuli (see [START_REF] Bernstein | Effects of an auditory signal on visual reaction time[END_REF], [START_REF] Radeau | The after-effects of ventriloquism[END_REF], [START_REF] Shams | Benefits of multisensory learning[END_REF] for example). Hence, online and progressive construction of sensory-motor representations is one of the current challenge of the developmental robotics field [START_REF] Kirstein | Towards autonomous bootstrapping for life-long learning categorization tasks[END_REF], [START_REF] Oudeyer | Developmental robotics[END_REF], [START_REF] Ridge | Self-supervised cross-modal online learning of basic object affordances for developmental robotic systems[END_REF] that targets the conception of robots that are able to learn to interact with an unknown environment in an autonomous and lifelong open-ended manner.

As an agent catches its internal state and the one of its surrounding environment through multiple sensors, sensorymotor representations have to be learned from the fusion of these various sensor data flows. Multimodal data fusion is a very complex task that raises challenging problems such as what, when and how to fuse data [START_REF] Atrey | Multimodal fusion for multimedia analysis: a survey[END_REF], even if databases are well prepared as in the machine learning field. Additionally, when learning multimodal data in the developmental robotic field, one has to consider that large areas of the sensorymotor space will never be learnable [START_REF] Baranès | R-iac: Robust intrinsically motivated exploration and active learning[END_REF]. One example among Mathieu Lefort and Alexander Gepperth are with the UIIS division, ENSTA ParisTech, 828 boulevard des Maréchaux, 91762 Palaiseau, France. Mathieu Lefort and Alexander Gepperth are members of Inria FLOWERS, Inria Bordeaux Sud-Ouest, 200 avenue de la vieille tour, 33405 Talence, France (email: {mathieu.lefort, alexander.gepperth}@ensta-paristech.fr).

many others is that it is hopeless to try to predict the olfactory consequence of an eye movement. In this article, we consider the problem of learning an input/output relationship where the input/output function is well defined, even if noisy, in some restricted areas of the input space -that we define as learnable areas -but is not existing in most of the space -that we defined as unlearnable areas. Learning in such partially learnable environments is a very challenging and crucial problem for the developmental robotic field.

To tackle this problem, we use an architecture that is cortically inspired. At a macroscopic level of description, the cortex is structured in interconnected functional areas at various levels, from low level sensory areas (V1, A1, ...) to high level areas in the frontal cortex. At a mesoscopic point of view, structure and function of these areas seem to be generic [START_REF] Holthoff | Functional mapping of cortical areas with optical imaging[END_REF], [START_REF] Kandel | Principles of neural science[END_REF], [START_REF] Miller | Processing in layer 4 of the neocortical circuit: new insights from visual and somatosensory cortex[END_REF]. Especially, self-organization (i.e. close neurons of a cortical area are sensible to close stimuli) is a widely spread computational paradigm that is mainly observed in low level sensory areas [START_REF] Bosking | Orientation selectivity and the arrangement of horizontal connections in tree shrew striate cortex[END_REF], [START_REF] Schreiner | Order and disorder in auditory cortical maps[END_REF], [START_REF] Wessinger | Tonotopy in human auditory cortex examined with functional magnetic resonance imaging[END_REF]. PROPRE (that stands for PROjection-PREdiction) is a generic architecture for multimodal representations learning [START_REF] Gepperth | Efficient online bootstrapping of sensory representations[END_REF]. Each data flow is projected on a self-organizing map, providing an internal representation of each current stimulus. From each representation, the system tries to predict the ones of all other data flows. A correct prediction can only be obtained if corresponding stimuli are correlated. A predictability measure, that depends on the ability of a representation to predict the other ones, modulates the corresponding projection learning so that to favor learning of predictable multimodal stimuli at the system level. In previous works, we already show that this predictability modulation provides better multimodal representations for real data classification [START_REF] Lefort | Multimodal space representation driven by self-evaluation of predictability[END_REF], where all data were labeled, and that these representations are able to adapt to changes in the input distribution [START_REF] Lefort | Propre: Projection and prediction for multimodal correlations learning. an application to pedestrians visual data discrimination[END_REF]. This article studies the PROPRE architecture facing the complex problem of partially learnable environments.

In the next section we introduce the PROPRE paradigm. In section III, we present the two partially learnable environments that we use. The first one is derived from the MNIST database [START_REF] Lecun | The mnist database of handwritten digits[END_REF] where we artificially corrupt some digit categories. The second one is a robotic arm direct model with a restricted visual field. We show that PROPRE provides local representations gathered in the learnable areas of the environments. This provides a simple way to discriminate between learnable and unlearnable parts of the environment and improves the performance of our model that learns the input/output relationship only where it exists. We discuss possible implications and extensions of our work in section IV.

II. PROPRE

A. Main paradigms

PROPRE is a generic neural paradigm for online learning of representations from multiple data flows. Its processing is based on the coupling of projection and prediction and its architecture is structured in three modules (see figure 1). The projection module uses the self-organizing map paradigm (SOM) to autonomously obtain a dedicated non linear topological projection of each unimodal input space. It thus provides a spatial representation of the current stimulus of each data flow. In the prediction module, each representation is used to linearly predict representations of all other data flows. Each prediction can only be accurate if the corresponding modal stimuli -through their low level internal representations -are correlated. A predictability module quantifies the quality of each prediction that represents the interestingness of the learned unimodal representation as it indirectly reflects the correlation between the multimodal stimuli. This predictability measure is used to modulate the corresponding projection learning so that to favor the representations of stimuli correlated across modalities thus providing an hybrid framework combining generative and discriminative learning. For more details about the PROPRE paradigm, please refer to [START_REF] Gepperth | Efficient online bootstrapping of sensory representations[END_REF]. In the multimodal architecture, the target data flow is the learned representations of other data flows. However, any target value can be used to learn any arbitrary input/output function (see figure 2) as we already did in [START_REF] Lefort | Propre: Projection and prediction for multimodal correlations learning. an application to pedestrians visual data discrimination[END_REF] for pedestrian pose orientation classification. In this article, we study this data flow processing when learning a partially learnable environment, i.e. that the output value does not depend on the input value for large areas of the input space. In this context, the predictability measure plays a crucial role for identifying learnable areas from the environment.

From a computational point of view, the reception of a stimulus in the model leads to one transmission and one learning step so that the model provides online learning (i.e. the stimulus is represented and learned at the same time). Technically speaking, the transmission stage consists on the feed-forward evaluation of each module activity (equations 1.a-b-c in figure 2). Then, the learning stage updates the weights of the plastic connections linking the modules (equations 2.a-b). The equations are detailed in the three next sections describing respectively each of the three modules of PROPRE.

B. Projection

The projection step consists on a classical Kohonen selforganizing map (SOM) [START_REF] Kohonen | Self-organized formation of topologically correct feature maps[END_REF] except for two points (see equation 2.a). Firstly, the neighborhood radius and the learning rate are constant over time, instead of decreasing, in order to have an online adaptation of the prototypes. Secondly, the prototype learning is modulated by the predictability measure to favor learning of representations that can predict the target value, which is the main feature of PROPRE.

In practice, S is a discrete bi-dimensional square grid of neurons that receives the input data flow D (see figure 2). The prototype at time t of a unit x in the map is denoted w SD (x, t). These prototypes are updated with the following equation:

∆w SD (x, t) = ηPr (t)e -||x-x * (t)|| 2 2 2σ 2 (D(t) -w SD (x, t))
(2.a) with η the constant learning rate, σ the constant variance of the Gaussian neighborhood radius, || • || 2 an euclidean distance in the map, Pr (t) the predictability measure (see section II-D) and x * (t) the best matching unit defined as the unit whose prototype is the closest to the current input

D(t), i.e. x * (t) = arg min x |w SD (x, t) -D(t)| 2 , with | • | 2
an euclidean distance on the input space.

The activity of any unit x of the map is defined as

S(x, t) = 1 if x = x * (t) 0 otherwise (1.a)
This simple map activation function provides a prediction that only depends on the best matching unit (see section II-C), so that there is no interference between units in the predictability measure computation (see section II-D). This allows to more precisely observe the influence of the predictability measure on the unit prototypes learning in partially learnable environments, which is the aim of this article. However, in order to obtain a better classification performance, other activation functions can be used [START_REF] Hecht | Using self-organizing maps for regression: the importance of the output function[END_REF].

C. Prediction

The projection activity S(t) is used to compute a prediction P(t) of the current value of the target data flow T(t) at each time t. The activity of a unit x in P is computed as a weighted sum of the S activity:

P (x, t) = y w P S (x, y, t)S(y, t) (1.b)
with w P S (x, y, t) the weight between the unit y in S to the unit x in P .

The weights of the connection between S and P are learned with a classical stochastic gradient descent implementation of a linear regression. This algorithm minimizes the mean square error between the prediction P(t) and the current target value T(t). Thus, the weights are updated with the following equation:

∆w P S (x, y, t) = η S(y, t)(T (x, t) -P (x, t)) (2.b)
with η the constant learning rate.

D. Predictability measure

The predictability measure modulates the projection learning depending on the quality Q(t) of the prediction P(t) with respect to the true target value T(t). It is defined as

Pr (t) = Q(t) -θ(x * (t), t) if Q(t) > θ(x * (t), t) 0 otherwise (1.c) Q(t) = P (z * , t) z P (z, t) with z * = arg max z T (z, t) θ(x, t) = (1 -τ )θ(x, t -1) + τ Q(t) if x = x * (t) θ(x, t -1) otherwise
In our experiments, the target value represents the category of the input, if any, so that the index of the highest target value corresponds to the input category, when it is defined (see section III). Thus, the quality of the prediction Q(t) represents the percentage of prediction of the right category. Please note that this measure can also be consistent with a regression learning task if the target value is encoded with a spatial coding, i.e. that the real value is represented by the spatial position of an activity in a map. This is precisely the case when PROPRE is used for multimodal learning [START_REF] Lefort | Multimodal space representation driven by self-evaluation of predictability[END_REF], as the target value is the representation of other data flows (see section II-A), which is the projection activity, here consisting on the best matching unit activated (see section II-B). θ(x, t) is a sliding threshold that is computed as the average prediction quality of a unit when it is the best matching unit (i.e. when x = x * (t)). Thus, it represents the average prediction quality in a local area of the input space, corresponding to the Voronoi cell of the unit. This sliding threshold is computed as an online average with a temporal horizon τ .

The predictability measure is equal to the difference between the current prediction quality and the average one of the current best matching unit 1 . Thus, the current stimulus is only learned if it provides a prediction locally more accurate than the average and the learning strength depends directly on this accuracy difference to the local average.

III. EXPERIMENTS AND RESULTS

For all our experiments, in order to quantify the influence of the predictability modulation on the system learning, we compare the PROPRE architecture described in section II with the same one without the modulation influence (corresponding to ∀t, Pr (t) = 1 in equation 2.a) that we denote SOM+LR in the following of the article. Unless specified otherwise, in section III-A (respectively III-B), all experiments are done with a 10 × 10 (respectively 20 × 20) SOM map and training occurs during 20 millions (respectively 100 millions) of time steps, where an input/output couple of values is randomly picked in the train dataset at each time step. Each initial connection weight is uniformly randomly chosen in [0, 1[, τ , η and η are set to 0.01.

A. MNIST database with corrupted digit categories 1) Protocol: We define several datasets derived from the classical machine learning MNIST database [START_REF] Lecun | The mnist database of handwritten digits[END_REF]. The handwritten digits pictures are used as the input data flow. The target value, which corresponds to the digit category, is defined for every picture in the MNIST database. In order to get partially learnable environments, we corrupt this target value for some digit categories. In practice, the target value is computed with the following steps (see figure 3):

• first, we corrupt some digits (1, 2, 4, 5 and 7 in our experiments) 2 by not providing the category to the system when a corresponding picture is presented as input

• second, distractors, i.e. other random categories, are added to the category vector (the number of distractors can be different for uncorrupted and corrupted digit categories)

• third, uniform noise is added to the category vector (the noise amplitude can be different for uncorrupted and corrupted digit categories)

So around half of the input space is unlearnable as the output does not depend on the input for 5 digit categories.

For our experiments we define 4 databases with various distractor numbers and noise amplitude, each time having around 90% of training data and 10% of test data (only containing uncorrupted digit categories) 3 : For uncorrupted digit categories -0, 3, 6, 8 and 9 in our experiments -(left column), the digit category is represented as in the MNIST database by a 1 at the right position in the category vector, but is missing for the other corrupted digit categories (right column). Some distractors, i.e. other values at 1, are added at random positions in the vector. Then, some uniform random noise with a fixed amplitude is added to all values of the vector, providing the output target value of the system (bottom).

• setup 1: 1 distractor is added to corrupted digit categories

• setup 2: random noise with amplitude 1 is added to corrupted digit categories

• setup 3: random noise with amplitude 1 is added to all digit categories

• setup 4: 1 (respectively 2) distractor(s) is (are) added to uncorrupted (respectively corrupted) digit categories and random noise with amplitude 0.1 is added to all digit categories Thus, setups 1 and 2 correspond to various kind of perturbations when the input/output relationship is not defined, whereas setups 3 and 4 also include perturbations of the input/output function where defined. Moreover, we define a fifth database as a baseline, where training data only contain uncorrupted digit categories without noise and distractors, denoted learnable environment in the following.

2) Learned representations: In figure 4, we present one example of the learned self-organizations by the projection map in the PROPRE and the SOM+LR architectures, that is representative of all four partially learnable setups. Without modulation, the projection map represents all the input space data, i.e. all 10 digits, in a self-organized way, which is the expected behavior of a Kohonen SOM. However, we can clearly observe that, with the predictability modulation, the SOM only represents digits from uncorrupted categories. Thus, thanks to the predictability measure modulation of the projection learning, the system autonomously learned to map the input space area where an input/output function is defined.

3) Classification performances: In the previous section, we show that the modulation introduced in PROPRE leads to representations that are gathered in the learnable area of the input space. As these representations are used as input of the prediction module, having a better mapping of the learnable area of the input space leads to a better classification performance of the system, which is shown in figure 5. Indeed, PROPRE is more efficient than SOM+LR in all setups, including the dataset where the input/output function is defined everywhere, confirming our previous results on other benchmarks [START_REF] Lefort | Propre: Projection and prediction for multimodal correlations learning. an application to pedestrians visual data discrimination[END_REF]. Moreover, the PROPRE performance on the partially learnable environments are close or even slightly better than the one of SOM+LR on the learnable environment. Thus, on this protocol, additionally to autonomously avoid unlearnable part of the environment, the predictability modulation is also able to improve the representation of the learnable area of the environment to get a better performance.

B. Motor direct model with restricted visual field 1) Protocol: We also tested our PROPRE architecture with some simulated robotic data, which is the targeted field of application of our model. We define several datasets corresponding to a motor direct model of a robotic planar arm with a restricted visual field (see figure 6). Thus, the robot is only able to see its hand in a limited area of its reachable space. The visual input of the robot is a simple 5 × 10 pixel matrix. If so (left), the corresponding pixel of the visual data is set to 1, otherwise no activity is provided (right). Some distractors and some uniform random noise with a fixed amplitude are added to the visual target data of the system (bottom).

Datasets are generated as following:

• an input motor command corresponding to

(θ 1 , θ 2 ) is randomly chosen in [0, π[×[0,

π[ with an uniform distribution

• if the hand is visible, a 1 is put at the corresponding position in the 5 × 10 pixel matrix

• some distractors, i.e. 1 values, are put at random position in the pixel matrix (the number of distractors can be different if the hand is visible or not)

• the visual target value is obtained by adding some random uniform noise with a fixed amplitude to the pixel matrix (noise amplitude can be different if the hand is visible or not)

Finally, the visual data only depends on the input motor command in around 20% of the reachable space, where the hand is visible. By the way, some part of the visual field are not reachable.

Similarly to what we did with the MNIST database, we generate 4 different datasets with various distractor number and noise amplitude, each time having 90% of training data and 10% of testing data (only containing motor inputs providing a visible hand position):

• setup 1: 1 distractor is added when the hand is not visible

• setup 2: random noise with amplitude 1 is added when the hand is not visible

• setup 3: random noise with amplitude 1 is added whenever the hand is visible or not

• setup 4: 1 (respectively 2) distractors are added when the hand is visible (respectively not visible) and random noise with amplitude 0.1 is added whenever the hand is visible or not Thus, setups 1 and 2 correspond to various kind of perturbations when the input/output relationship is not defined, whereas setups 3 and 4 also include perturbation of the input/output function where defined. Moreover, we define a fifth database as a baseline, where training data contain only motor commands leading to a visible hand, denoted learnable environment in the following.

2) Learned representations: We can clearly see in figure 7 the same effect that we observed with the MNIST database. Indeed, without modulation of the projection learning, the prototypes represent all the input space in a generative way whereas, with the modulation by the predictability measure, prototypes are gathered mainly in the learnable area of the input space. It has to be noted that, with all initial values between 0 and 1, prototypes are initialized far away from the learnable area of the input space.

Another way to study this gathering of prototypes in the learnable area of the input space is to use the distance to the best matching unit to build a binary classifier between learnable and unlearnable areas of the environment. Thus, once the model has converged, we classify the input data of a testing dataset (here including also data from the unlearnable area) as coming from the unlearnable area (respectively learnable area) if the distance between the input and the best matching unit is above (respectively below) some threshold. The corresponding ROC curves of these binary classifiers when using prototypes learned by SOM+LR and PROPRE are represented in figure 8.

We can clearly see that the classifier based on the prototypes of PROPRE is largely better than the one built with the prototypes learned without the predictability modulation. This confirms that PROPRE can autonomously learn local representations in the learnable area of the environment.

3) Classification performances: We also trained a multilayer perceptron (MLP) on all setups to have a baseline of performances obtained by a standard machine learning discriminative algorithm. The hidden layer of the MLP was set to 400 units so that the internal representation in all models have the same size. Performances obtained by the various models on the different setups are shown on figure 9. A prediction error is counted when the maximum of the prediction does not correspond to the real position of the hand in the visual field. Fig. 7. Top (respectively bottom): examples of prototypes learned by a 10 × 10 self-organizing map, represented in the motor input space, when no modulation (respectively predictability modulation) is applied (same qualitative results are obtained on all four partially learnable setups). The gray area represents the motor space leading to a visible hand, i.e. the learnable area of the input space.

When trained only on the learnable area of the input space, all models (SOM+LR, PROPRE and MLP) achieve logically their best classification performance. In this case, the best performance is obtained by the MLP, as expected, as it is a discriminative learning algorithm. By the way, PROPRE performance is, once again, better than the one of SOM+LR. The more interesting results concern the 4 setups where a large area of the input space is unlearnable, that does not fit with classical machine learning framework. In these cases, the SOM+LR performances decrease dramatically, as the SOM represents the entire input space (see previous section). Moreover, the MLP performances also decrease and are worst than the one of PROPRE. By the way, Hartono and colleagues proposed a model whose equations are close to the ones of PROPRE but are derived from a stochastic gradient descent so that the model is a discriminative learning algorithm [START_REF] Hartono | Classificability-regulated selforganizing map using restricted rbf[END_REF]. Preliminary results seem to indicate that this model has the same qualitative behavior as a MLP, i.e. that its performance is deeply impacted by unlearnable areas. This illustrates the main interesting monitoring property of the PROPRE predictability module to detect and gather Fig. 8. Top (respectively bottom): ROC curves of a classifier that classifies data as coming from the unlearnable or learnable area of the input space depending on the distance to the best matching unit of the SOM learned with SOM+LR (respectively PROPRE). representations in the learnable area of the input space. Thus, PROPRE only tries to regress the input/output function in the learnable area and is quite insensible to the existence of unlearnable areas contrary to classical discriminative learning approaches.

4) Robustness to instabilities in the learnable area: In all previously presented experiments, the input/output function, even if noisy, was always present in the learnable area of the input space and never defined elsewhere. This is precisely what distinguishes learnable from unlearnable areas. In this section, we study the robustness of PROPRE learning to some missing target values in the learnable area to be more representative of real data. For that purpose, we take the setup 2 (see section III-B1) and, with some probability, suppress the visual location of the hand in the pixel matrix when the hand should be visible. This may correspond to the occlusion of the hand behind an object for example. Performances obtained with PROPRE and SOM+LR architectures on this setup are presented in figure 10.

We can see that, for both models, the performance decreases with the increase of the percentage of input/output relationship instability in the learnable area as the prediction step has more difficulties to get a relevant statistic in the data. Interestingly, PROPRE performance is only gradually degraded with the instability percentage and always achieves a better performance than SOM+LR in all cases. By the way, even with up to 30% of instability, PROPRE performance remains better than the one of SOM+LR on the data without any instability.

IV. DISCUSSION AND PERSPECTIVES

PROPRE is a generic and modular paradigm whose architecture and processing are inspired by the ones of the cortex. It provides an input/output relationship online learning based on the combination of projection and prediction, that is well suited for multimodal learning. The input data flow is projected on a learned self-organizing map that provides a low dimensional of the current input. This internal representation is used to predict the value of the output target. The main originality of PROPRE is the use of a predictability measure, that depends on the monitoring of the prediction quality, to modulate the projection learning. This predictability modulation leads to learning of representations that are locally useful for the global output target prediction.

PROPRE was already successfully applied to unimodal Fig. 10. Top (respectively bottom): temporal evolution of the performance of SOM+LR (respectively PROPRE) when learning on setup 2 depending on the percentage of input in the learnable area of the environment whose target value is missing (refer to the text for more details). and multimodal learning in real data classification tasks [START_REF] Lefort | Propre: Projection and prediction for multimodal correlations learning. an application to pedestrians visual data discrimination[END_REF], [START_REF] Lefort | Multimodal space representation driven by self-evaluation of predictability[END_REF], where all data are labeled. In this article, we study the PROPRE paradigm when learning environments that include a large unlearnable area in the input space, i.e. where the output value does not depend on the input. This kind of problem is not typical of the machine learning field, benchmarks assume an hidden and noisy input/output function everywhere. However, this very challenging problem is crucial for the robotic developmental field. Indeed, robots face high dimensional sensory-motor environments, where large part of these environments cannot be predicted at any time of the development and is thus hopeless to try to learn [START_REF] Baranès | R-iac: Robust intrinsically motivated exploration and active learning[END_REF].

We construct multiple benchmarks of partially learnable environments with various random uniform noise and distractors in the output target value. Firstly, we use the MNIST database where we artificially suppress the category for half of the digits. Secondly, we simulate a robotic arm in a plan, whose hand can be seen in a restricted visual field only covering around 20% of the reachable space.

The modulation mechanism introduced in PROPRE leads to a gathering of the local representations in the learnable areas of the input space. So, as PROPRE regresses the output value from these internal representations, it only learns the input/output relationship where it is defined. Thus, when some areas of the input space are unlearnable, PROPRE performances are better than the ones achieved by the same architecture without predictability modulation and are also better than the ones of a multi-layer perceptron, a standard discriminative learning algorithm, in the robotic setup. On the contrary, the MLP obtains the best performance when the input/output relationship is defined on all the input space. This validates the important role of the predictability module to be able to learn in partially learnable environments by detecting learnable areas from unlearnable ones. This property may also be used to build a classifier that discriminates inputs from the learnable areas in the input space based on the distance between the input and the best matching unit as we illustrate it on the direct arm model learning. By the way, PROPRE performance is only slightly degraded by adding noise to the input/output function where it exists and is gradually degraded when the input/output function is missing for some increasing percentage of inputs of the learnable area, which are both interesting robustness properties.

The successful application of PROPRE to learning in partially learnable environments, lies on the monitoring provided by the predictability modulation. It is based on the prediction quality measure that is defined as the percentage of prediction of the right category. As mentioned in section II-D, it can also be applied to a regression task as we already did it in [START_REF] Lefort | Multimodal space representation driven by self-evaluation of predictability[END_REF] for multimodal learning. Thus, a direct perspective of our work is to study PROPRE facing real multimodal data with a partial relationship between the modalities, which is particularly relevant as each sensor usually has its own characteristics. Moreover, preliminary results indicate that this predictability measure can be extended to a multiple categories representation, with some constraints, even if more work is needed to on this aspect.

The discrimination between learnable and unlearnable areas of the environment, that is autonomously learned by PROPRE, can be very interesting from a developmental robotics perspective. Even if we show in this article that PROPRE can deal with partially learnable environments, exploration of highly dimensional sensory-motor environments cannot be extensive within a live time. Thus, active exploration mechanisms have to be used for guiding exploration towards interesting sensory-motor areas, that are defined as the one leading to competence progress in the artificial intrinsic motivation framework [START_REF] Oudeyer | Intrinsic motivation systems for autonomous mental development[END_REF] for example. It may be interesting to study the coupling of PROPRE with such intrinsic motivation framework as the autonomous detection of learnable areas of the input space must provide a great help for active exploration.

Fig. 1 .

 1 Fig.1. PROPRE architecture is structured in three interacting modules. A projection module learns a low dimensional representation of each modal stimulus. A prediction module provides, from each unimodal representation, a prediction of all other representations. A predictability measure quantifies the quality of each prediction and modulates the corresponding projection learning.

Fig. 2 .

 2 Fig. 2. One data flow processing in PROPRE. The system learns internal representations of the data flow that help the learning of the input (D)/output (T ) relationship.

Fig. 3 .

 3 Fig. 3. Input data are the handwritten digits of the MNIST database (top).For uncorrupted digit categories -0, 3, 6, 8 and 9 in our experiments -(left column), the digit category is represented as in the MNIST database by a 1 at the right position in the category vector, but is missing for the other corrupted digit categories (right column). Some distractors, i.e. other values at 1, are added at random positions in the vector. Then, some uniform random noise with a fixed amplitude is added to all values of the vector, providing the output target value of the system (bottom).

Fig. 4 .

 4 Fig. 4. Top (respectively bottom): prototypes learned by the projection map when no modulation (respectively predictability modulation) is applied. Each square represents the prototype of the unit at the corresponding position in the 10 × 10 SOM.

Fig. 5 .

 5 Fig. 5. Final classification errors on the different setups (refer to section III-A1 for details) depending on the use of the predictability modulation.

Fig. 6 .

 6 Fig.6. A robot moves its arm (d 1 = 3 and d 2 = 2.5) in a plan depending on an input joint motor command (top). It can only see its hand in a restricted visual field. If so (left), the corresponding pixel of the visual data is set to 1, otherwise no activity is provided (right). Some distractors and some uniform random noise with a fixed amplitude are added to the visual target data of the system (bottom).

Fig. 9 .

 9 Fig.[START_REF] Kandel | Principles of neural science[END_REF]. Top (respectively bottom): temporal evolution of classification errors on the different setups (refer to section III-B1 for details) when no modulation (respectively predictability modulation) is used. On the bottom figure, dashed lines represent the final performance obtained by a multi-layer perceptron, the color indicates the setup on which the MLP was trained.

This is a slight adaptation of our previous algorithm, where predictability value was binary (learning on or off)[START_REF] Lefort | Multimodal space representation driven by self-evaluation of predictability[END_REF], that allows more precise modulation.

We initially choose 0,

3, 6, 8 and 9 as uncorrupted digit categories as the digits are visually similar so that having more prototypes for representing these digits may improve more clearly the classification performance. However, tests on other configurations show that the results are qualitatively independent of this ad hoc choice.[START_REF] Bernstein | Effects of an auditory signal on visual reaction time[END_REF] In practice, we split the MNIST database in 80% of training data and 20% of test data. However, as we only test our model on uncorrupted digit categories, we only keep around half of the initial testing dataset.

ACKNOWLEDGMENT

The authors want to thank Thomas Hecht for his useful remarks and reviews.