CHALLENGING PLATONIC ERÔS THE ROLE OF THUMOS AND PHILOTIMIA IN LOVE

Plato's Dialogues continue to challenge our common definition of love. Whereas we commonly understand the primary meaning of love as an intense feeling addressed to an individual and attachment to his uniqueness -what we may call "personal love" hereafter -, Plato famously defines erôs as a force that drives each individual towards an object that appears to be rather abstract: Beauty (kalon). Platonic love provides the individual with happiness and leads him to virtue by knowledge of the Good. This intellectualized form of love goes together with a condemnation of sexual attachment and greediness, starting with the Symposium and the Phaedrus through to the Laws. It is therefore commonly argued that, for Plato, "personal love" cannot be an end in itself. As Vlastos has shown, love for an individual is a mere symptom of human deficiency, for the only thing truly worth loving is the Good, in a rather egoistic way. 1 It is then a small step from the condemnation of inferior love whose paradigm is sexual attachment to an overall rejection of "personal love", insofar as the lovers have not fulfilled the process of ascending the so-called "ladder of love" depicted in the Symposium (210a-212a). 2 Real Platonic love, the only one of value, is the philosophical love for the Good. 3 What remains unclear, though, is whether real philosophical love represents a distinct kind of love, which has nothing to do with inferior kinds. 4 If this were the case, how would a nonphilosopher understand that philosophical love is true erôs? And, if it is even possible, how could inferior kinds of love be used as means of experiencing love of Forms, and, even more so, the Form of the Beautiful (kalon)? 5 A way of rehabilitating "personal love" in Plato, and of organizing different kinds of love into a non-exclusive hierarchy, is to follow Socrates when he decides on a typology of erôs at the end of the Republic. As Socrates shows in Book IX of the Republic, there are three main types of desires in the human soul: bodily desires (thirst, hunger, sexual desires, and money as a means of buying whatever one desires), desire I'm grateful to the audience of the conference "Erôs in Ancient Greece", and especially Ed Sanders and Chiara Thumiger and Lee Brooks for their helpful remarks and objections.

1 Vlastos (1981). 2 This opposition between an exclusive platonic love and a more common one has to be qualified. For an example of how love can be more inclusive, allowing sexual intercourse for instance, see Gill (this volume) on the stoic interpretation of platonic love. 3 Vlastos's ground-breaking article has been much discussed. See first Nussbaum (2001), who opposes the Symposium to the Phaedrus (ch. 6 and 7) on the grounds that, in the latter, Plato acknowledges the importance of sensibility, passions and personality in the experience of love (p. 213-23). Price (1981) maintains, more accurately, that the conception of love in the Symposium is an "inclusive" one (the philosophical love contains the others), whereas the Phaedrus ends by discarding inferior forms of love, symbolized by the desires of the two horses. See also the critical notes by Rowe (1990). White (1990) shows in what sense "personal love" could fit the Platonic picture of philosophical love. Finally, see Gill (1990), who shows that the myth aims at educating the lovers to respect and esteem each other (p. 76-78) but that it is not an end in itself; the main goal is to unify one's soul in experiencing philosophical love, i.e. by submitting the desires to the rule of reason (p. 82ff.). 4 For a detailed account of Plato's strategy in transforming the common conception of love in the Phaedrus, see Cairns (this volume).

for honour and victory, and desire for knowledge. These three types of desire define three corresponding types of man: the profit-loving man (philochrêmatos), the victory and honourloving man (philonikos kai philotimos) and the philosopher (philosophos) (Resp. IX, 580d-581b). The opposition between the first and the last character reaches its climax when Socrates depicts the similarities and differences between the violent erôs growing in the tyrant's soul, and the erôs of the philosopher. 6 Less attention has been paid to the other kind of love, philotimia, which arises not from the desiring part of the soul but from its intermediary part: thumos. The philotimos seems at first sight to be an unfamiliar image of a lover, but he is actually the only one who feels what we can recognize as a "personal" attachment to somebody, as well as care for the reputation and the image of his lover and of himself. Indeed, and contrary to the appetitive type of love, thumoeidic love as described in the Phaedrus is not easily discarded, insofar as a smoother treatment seems to be reserved to the thumoeidic lovers (Phdr. 256b-e). 7 This chapter aims to show that philotimia is seen by Plato both as a dangerous challenge to philosophical love and also as an opportunity to use the energy expended by the psychic function from which this love comes, thumos, in order to serve philosophical goals. I shall begin with an analysis of the role of philotimia in the speeches of Phaedrus and Diotima in the Symposium. Then, I shall explain, on the grounds of the tripartite soul in the Republic, how philotimia can be used as a lever for philosophical erôs in the Phaedrus, before concluding on its political use.

Philotimia as a wrong basis for love

In the first praise speech of the Symposium, Phaedrus makes philotimia the sense of honour and shame, the most important lever of erôs, and the instrument of our own flourishing and self-construction through others. 8 I cannot say what greater good there is for a young boy than a gentle lover, or for a lover than a boy to love. There is a certain guidance each person needs for his whole life, if he is to live well; and nothing imparts this guidance -not high kinship, not public honor, not wealth -nothing imparts this guidance as well as Love (οὔτε συγγένεια οἵα τε ἐµποιεῖν οὕτω καλῶς οὔτε τιµαὶ οὔτε πλοῦτος οὔτ΄ ἄλλο οὐδὲν ὡς ἔρως). What guidance do I mean? I mean a sense of shame at acting shamefully, and a sense of pride in acting well (τὴν ἐπὶ µὲν τοῖς αἰσχροῖς αἰσχύνην͵ ἐπὶ δὲ τοῖς καλοῖς φιλοτιµίαν). Without these, nothing fine or great can be accomplished, in public or in private. What I say is this: if a man in love is found doing something shameful, or accepting shameful treatment because he is a coward and makes no defense, then nothing would give him more pain than being seen by the boy he loves -not even being seen by his father or his comrades. We see the same thing also in the boy he loves, that he is especially ashamed before his lover when he is caught in something shameful (Symp. 178c3-e3, trans. Nehamas and Woodruf) 9 For Phaedrus, erôs is the foundation of ethics. Through shame and philotimia, one becomes better, and better able to care for oneself. Phaedrus compares erôs to three other types of social relationships: philia (especially between members of the same family or genos), honour-based relationships (which may refer to a more extended sphere of social and political relationships), and wealth-based relationships. Faced with these three relationships, erôs remains the only one that leads the individual to real happiness (eudaimonia). According to Phaedrus, the sense of honour enables the lover to take up the challenge of showing grace, generosity and magnanimity in front of his lover. It is tempting to say that Phaedrus's speech is a mere echo of popular morality and expresses concern for what people say. The value of the individual's behaviour and actions would then depend on the way witnesses evaluate it, according to existing social norms. There would only be a difference in degree between the shame experienced in front of the beloved and in front of any other individual. But this interpretation is misleading. The rule of construction of selfhood is assuredly heteronomous, insofar as selfhood depends on how a particular witness, the lover, considers it. But this does not mean that the values advocated by the parents, the philoi, and others in general, have the same function as the ones implied in a relation of love. Because the lover cares for his beloved far more than he does for his parents, friends and others, the beloved functions as a real motivation for the lover to show an ideal image of himself. For Phaedrus, absolute timê is achieved when one is ready to sacrifice his life, being entirely at the lover's behest. 10 In experiencing shame and philotimia in front of the beloved, one commits oneself to values, which one could have otherwise only superficially approved. In other words, philotimia and shame are, according to Phaedrus, means for the individual to experience the values he is committed to as true ones, and to seek new challenges for himself. However, it seems at first sight that Phaedrus is not really consistent in opposing a sense of honour and shame, which comes from erôs, and the attachment to social and political marks of honour (timai). There seems to be a contradiction in the fact that Phaedrus first points out the inanity of founding happiness on timai, and then makes philotimia the cause of becoming better. But we should assume here that Phaedrus gives the concept of timê different meanings. 11 In the first case, timê seems to refer to a kind of possession (civic honours, prerogatives, or even presents), whereas the latter, the one philotimia strives for, does not mean the product of the relation, but the value of the relation itself, so that selfhood is constructed through a sense of honour and shame, placing the other on a level beyond the possessions one can get from him. Phaedrus's speech plays with a Homeric legacy. The persistent Homeric patterns, notably the mention of the hero's spirited force (menos) (178e3-179b3), and the use of the figure of Achilles as the paragon of courage (179e1-180b5), grace and sensibility at the end of the speech, are important clues of a link between erôs and philotimia, and even with their psychological seat, thumos. If we go back to Homeric epics, erôs is likely to be found in the 10 Self-sacrifice could first be thought to be an expression of perfect philia, as Aristotle states in Eth. Nic., IX.8, 1169a18 ff., where he employs the same verb (ὑπεραποθνῄσκω) to show how philoi are ready to die for each other (I am grateful to Ed Sanders for drawing this passage to my attention), as Phaedrus to describe the sacrifice of Alcestis and Achilles (Symp. 179b4, 180a1). Whether or not Aristotle has the Symposium in mind, it underlines the oddity of Phaedrus's speaking of erôs when we expect philia. However, it might also mean that for Phaedrus, erôs is even more inclusive than philia, for a perfect lover such as Achilles not only "dies for" Patrocles but "follows him in death" or "dies after" him (οὐ µόνον ὑπεραποθανεῖν ἀλλὰ καὶ ἐπαποθανεῖν τετελευτηκότι, 180a1). character's thumos, together with philia and passions of the like. 12 Furthermore, it is in their thumos that heroes feel anger, hatred or delight when watching their enemy perish. 13 Phaedrus is thus recalling a traditional link between erôs and thumos, the latter being the aggressive side of the former. 14 Furthermore, it is interesting to note that Phaedrus' speech mentions all the characteristics of the thumos that will be found in the Republic: an acute sensitiveness to what is good and just through a sense of honour and shame; marks of manliness; a spontaneous manner of showing one's commitment to the values one believes in; a propensity to philotimia and, more generally, to a competitive spirit, leading one to seek out new challenges. It is likely that in the Republic, Plato does not discard this Homeric legacy concerning the sensitivity to timê, which is rooted in the thumos of the auxiliaries.

Before we turn to the main criticism Diotima will make in her discourse against this thumoeidic ideology, we should compare this speech to its "echo" in Plato's dialogues, namely Lysias's speech in the Phaedrus. When reading Phaedrus's speech in the Symposium, we could have expected the word philia to occur instead of that of erôs, as if Plato were purposely transposing a Homeric model of personal relationships to fit artificially with erôs. As a counterpoint to Phaedrus's speech, Lysias's discourse in the Phaedrus, read by the same Phaedrus he is in love with, employs the same notions of shame and sense of honour as virtues.15 But Phaedrus's speech is reversed, for Lysias praises the non-lover, and criticizes the incontinence and hybris of the lover. Thus, Lysias dissociates philotimia from aischunê, the first being related to envy and jealousy. Now suppose you're afraid of conventional standards (τὸν νόµον τὸν καθεστηκότα δέδοικας) and the stigma that will come to you if people find out about this. Well, it stands to reason that a lover -thinking that everyone else will admire him for his success as much as he admires himself (οἰοµένους καὶ ὑπὸ τῶν ἄλλων ζηλοῦσθαι ὥσπερ αὐτοὺς ὑφ΄ αὑτῶν) -will fly into words and proudly declare to all and sundry that his labors were not in vain (καὶ φιλοτιµουµένους ἐπιδείκνυσθαι πρὸς ἅπαντας). Someone who does not love you, on the other hand, can control himself and will choose to do what is best, rather than seek the glory that comes from popular reputation (ἀντὶ τῆς δόξης τῆς παρὰ τῶν ἀνθρώπων)(Phdr. 231e3-232a6, trans. Nehamas and Woodruf).

Unlike the lover, who is moved by philotimia, the non-lover is able to show reserve: 12 Numerous formulae indicate that thumos is that by which we feel a certain attachment to something. Such feelings range from kindness and philia between comrades, friends, or members of the family (Il. I, 196, 209; V, 243, 826; IX, 486; X, 531; XI, 520, 608; XIX, 287; XXIV, 236, 748, 762; Od. VI, 20; XIV, 146) to mere sexual attraction as in the episode of the seduction of Zeus (Il. XIV, 315) or enchantment as in the appearance of Penelope in front of her suitors (Od. XVIII, 212, 282), through to what we can call "love" between husband and wife for example, including both philia and a sexual relationship (Il. IX, 340-343, 398; Od. XVII, 553-555). 13 See for instance Il. I,196;I,209;I,562;V,243;V,826;IX,341;IX,398;IX,486;XIV,315;XIX,287;XXIII,548;XXIII,595;XXIV,762;Od. VI,23;XIV,146;XV,20;XVIII,212. 14 Hesiod,, where we are told that Philotês is born at the same time as Nemesis, and Eris which affect the thumos (karterothumos). See also Sappho, fr.1 (West), where the poetess prays to the Divinity to strengthen her thumos, becoming the active, or even aggressive side of her deceitful erôs, whether her thumos is defeated by nauseous disease (l. 4), or succeeds in conquering her love (l. 18 and l. 27).

[No, it's proper, I suppose, to grant your favours] not to people who achieve their goal and then boast about it in public (οὐδὲ οἳ διαπραξάµενοι πρὸς τοὺς ἄλλους φιλοτιµήσονται), but to those who will keep a modest silence with everyone (ἀλλ' οἵτινες αἰσχυνόµενοι πρὸς ἅπαντας σιωπήσονται)(Phdr. 231e3-232a6, trans. Nehamas and Woodruf).

In opposing Phaedrus and Lysias's discourses, Plato aims to criticize philotimia as a good motivation in erotic relationships. Later in the Phaedrus, giving his own version of Lysias's speech, Socrates playfully recognizes in the supposed non-lover a lover necessarily prone to jealousy (φθονερὸν δὴ ἀνάγκη εἶναι, 239a7-b1). Moreover, the lover would envy his beloved's wealth and rejoice in seeing it all scattered (ἐξ ὧν πᾶσα ἀνάγκη ἐραστὴν παιδικοῖς φθονεῖν µὲν οὐσίαν κεκτηµένοις͵ ἀπολλυµένης δὲ χαίρειν, 240a5-6). As a result of these multiple reversals, it can be argued that for Plato, and contrary to what Phaedrus and Lysias say, an erotic relationship which relies on either one or both philotimia and aidôs, is doomed to failure because of its attachment to an image of oneself, which is dependent on what others say. Phthonos will necessarily appear, transforming the erotic relationship into a theatrical scene, where the self is but an image of what his beloved wants him to be and, reciprocally, the beloved a potential opponent in the competition that leads to happiness. 16 In response to Phaedrus' speech, Diotima's task consists in assuming the natural origin of philotimia in love without letting it fix the object of love on timê.

"Be sure of it, Socrates. Look, if you will, at how human beings seek honour (τὴν φιλοτιµίαν). You'd be amazed at their irrationality (θαυµάζοις ἂν τῆς ἀλογίας), if you didn't have in mind what I spoke about and if you hadn't pondered the awful state of love they're in, wanting to become famous and "to lay up glory immortal forever" (ἐνθυµηθεὶς ὡς δεινῶς διάκεινται ἔρωτι τοῦ ὀνοµαστοὶ γενέσθαι καὶ κλέος ἐς τὸν ἀεὶ χρόνον ἀθάνατον καταθέσθαι), and how they're ready to brave any danger for the sake of this, much more than they are for their children; and they are prepared to spend money, suffer through all sorts of ordeals, and even die for the sake of glory. Do you really think that Alcestis would have died for Admetus," she asked, "or that Achilles would have died after Patroclus, or that your Codrus would have died so as to preserve the throne for his sons if they hadn't expected the memory of their virtue -which we still hold in honor -to be immortal (µὴ οἰοµένους ἀθάνατον µνήµην ἀρετῆς πέρι ἑαυτῶν ἔσεσθαι)? Far from it," she said. "I believe that anyone will do anything for the sake of immortal virtue and the glorious fame that follows (τοιαύτης δόξης εὐκλεοῦς); and the better the people, the more they will do, for they are all in love with immortality (τοῦ γὰρ ἀθανάτου ἐρῶσιν)." (Symp. 208c1-e1, trans. Nehamas and Woodruf) Phaedrus's edifying conception of love according to which the lover seems to be entirely at the disposal of his beloved is refuted by a psychological analysis of the lover's real motives. Philotimia is a contradictory desire: through self-sacrifice, the lover clearly seeks a greater good: to die for somebody is nothing else than to die as a hero, i.e. overcome death by gaining immortality in renown. Diotima uses Phaedrus's own vocabulary and examples in order to deny that the object of love could be anything other than glory and fame. Using the example of Codrus, whose motivations are not erotic but political, Diotima denies that the role of the beloved is more important than the struggle for timê.

A twofold conclusion can be inferred from Diotima's speech: Phaedrus is right in saying that erôs is the foundation of ethics and of a set of values that leads to the Good, but he is wrong in making philotimia its achievement. Philotimia is an irrational desire (ἄλογος), for it leads the individual to prefer death as a way to acquire renown through immortality. Philotimia is then dangerous for two reasons: first, it transforms love into an edifying affair, exalting dramatic emotions; second, it fails to understand what is really, in love, the object to be valued. The philotimos focuses selfishly on the capacity of others to make him good, beautiful and virtuous. In other words, Plato denounces the propensity of the philotimos to love being praised at any cost, regardless of the real and objective value of actions and behaviour. Thus, for Plato, the philotimos remains dependent on what people think and say (through memory and renown) and fails to recognize what the real object of erôs is. However, Diotima does not entirely dismiss philotimia as a whole. As irrational as philotimia may be, it remains an expression of a longing for a real object of immortality. Contrary to Lysias in the Phaedrus when he criticizes cupidity and lust, or contrary to Socrates in Book VIII of the Republic, Diotima rightly points out the efficiency of philotimia as a lever for virtuous behaviour. Indeed, the three examples of Alcestis, Achilles and Codros are illustrious ones; what is therefore questioned is not the effect of philotimia but the way those heroes have misconstrued the image of the object of love. What is at stake, then, is how the psychological force of philotimia could be a lever to attain real immortality.

Educating thumos's force

If the Symposium presents the inferior type of love experienced by the philotimos as ambiguous, it does not explain how it can be used as a means towards attaining true philosophic love. The tripartite model of the soul might throw light on how philotimia can be diverted to this end. 17 In the Symposium, erôs is the name Plato gives to the fundamental desire which leads every single animal in the sensible world to seek its own good during its life, but the tripartite model can also be used in order to show both how different types of love could be classified on a scale of values, whose criterion would be their proximity to philosophy, and how philosophical love could proceed from a relatively inferior type of love. The link between the two intermediaries is philotimia. 18 In Book IV of the Republic, Socrates presents what is called the tripartite psyche. The psyche is divided into three "parts" or rather three "functions", each of which is responsible for a kind of action or passion: the rational part (logistikon), the desiring part (epithumêtikon), and the intermediary part (thumoeides). 19 The function of thumos is to value things, not evaluate them cognitively (which is the task of the logistikon, the rational part of the soul), nor desire them (which is the concern of the epithumêtikon). 20 Angela Hobbs has brilliantly summed up its function, saying that "the essence of the human thumos is the need to believe that one counts for something, and that central to this need will be a tendency to form an ideal image of oneself in accordance with one's conception of the fine and noble". 21 Valuing things means for each individual to be committed to his own values (laws, habits, norms), which will arouse two opposite emotions in him: anger (orgê, or sometimes thumos) when these values are threatened or despised (440c7-d6), and shame (aidôs or aischunê) when he himself fails to enact them (439e6-440a3, 440c1-c6). Central to the definition of the function of thumos is the idea of honour (timê). As we've seen in Phaedrus's speech in the Symposium, the greek word "timê" can refer to particular objects such as civic honour, special prerogatives given by public esteem, presents, etc. These objects are the expression of a complex network of social and political relationships, often referred to as the ideology of honour. 22 In this network, one is able to form oneself through others, what they think and what they say (see esp. 364a6-b2, 413e5-414a4). But timê also means simply the "value" the individual gives to something, which can be, potentially, anything. 23 Timê refers then to the way thumos gives something a value into [in] a particular context. This complex operation of valuing things makes timê ambivalent: it refers both to the act of "valuing something" and to an "object" which is actually the result of the process of valuing things in a given society which has its own set of values. 24 In that respect, thumos's force is at once a threat and a potential auxiliary to reason's rule.

In the Phaedrus the tripartite model of the soul is a means used by Plato to neutralize philotimia, focusing on its seat, thumos. The soul's partitioning enables Plato to locate the causes of love and act on them in order to transfigure them. A close reading of the central part of the charioteer's myth in the Phaedrus shows that thumos is actually the seat of many erotic effects 25 . It is then necessary to address a discourse to thumos in order to persuade it to care for the rational part of the soul, the "true self", represented by the coachman. Plato keeps thumos as a mover, but neutralizes its propensity to corrupt intellectual love. In the last part of the myth, Socrates depicts the effects of love on the lover using a physiological vocabulary that is often related to thumos: 20 The interpretation of the tripartite soul is, of course, a much-debated issue. Here I take as given a minimal reading of the tripartite model, as an intentional trifunctional structure, in which each function (reason, thumos and appetite) is capable of only one kind of operation. Reason is a calculative function, appetite is a conative force, and thumos is a "valuing function". Whereas book IV of the Republic presents each function as an autonomous agent, they are actually parts of an entire psychic structure in books VIII and IX. See, if I may, Renaut (2005) for such a reading, along with the relevant bibliography.

21 Hobbs (2000) 30.

22 Such an ideology is severely condemned by Socrates in the Republic. According to Socrates, timê, considered through its object, inevitably comes to focus on money, as if philotimia were only a mask for cupidity and the striving for wealth. The explanation runs as follows: in his quest for honour, one is surreptitiously driven to seek the means of acquiring it; as wealth could appear as a symbol of honour and public esteem, a philotimos could be easily corrupted into a philochrêmatos. See Rsp. 345a3-6, 390e9 where Socrates condemns Achilles for desiring a ransom, 547b7-549b9 where Socrates analyses how the philotimos is corrupted by love of money, and 550c11-553d7, which depicts the oligarchic man who only values money. 23 See for example the democratic man, who considers that all things have the same "value" (561b8-c4 ). 24 Many passages in the Republic compare the real and objective values with false ones: 336e8 on justice compared with gold, 485b6-8 on Forms compared to traditional objects of enquiry, 509a4-5 on the supreme value of the Good, 591b4-7 on the soul compared to the body, 595c2-3 on the respect which is due to truth compared to the respect due to a man. 25 As Cairns (1993) 384-5 has shown.

Once he has looked at him, his chill (ἐκ τῆς φρίκης) gives way to sweating and a high fever comes over him, with sweat and unwonted heat (ἱδρὼς καὶ θερµότης ἀήθης); (…) Now the whole soul seethes (ζεῖ οὖν ἐν τούτῳ ὅλη) and throbs (ἀνακηκίει) in this condition. Like a child whose teeth are just starting to grow in, and its gums are all aching and itching (κνῆσίς τε καὶ ἀγανάκτησις) -that is exactly how the soul feels when it begins to grow wings. It swells up and aches and tingles as it grows them (ζεῖ τε καὶ ἀγανακτεῖ καὶ γαργαλίζεται φύουσα τὰ πτερά) (Phdr. 251a7-c5, trans. Nehamas and Woodruf).

In this description, it is the whole soul that endures the affections of love such as sweating, blushing, throbbing, and so on. But in other passages of the dialogues, these symptoms refer to the state of the thumos. 26 These symptoms are more particularly to the emotion of shame. 27 Here, Socrates seems to speak by way of metonymy, using the word "soul" to refer to "thumos", which is merely of part of the soul. Why so? Certainly because the feelings aroused by the encounter between the lover and the beloved are always mixed, never reducible to sexual or intellectual feelings. It is then the thumos which is at stake here, for it is an emotional centre from which arises, after reflection, two other poles, epithymêtikon and logistikon. As the passage continues, Socrates proposes a more precise explanation:

Now when the charioteer looks in the eye of love, his entire soul is suffused with a sense of warmth and starts to fill with tingles and the goading of desire. As for the horses, the one who is obedient to the charioteer is still controlled, then as always, by its sense of shame, and so prevents itself from jumping on the boy (ὁ µὲν εὐπειθὴς τῷ ἡνιόχῳ τῶν ἵππων͵ ἀεί τε καὶ τότε αἰδοῖ βιαζόµενος͵ ἑαυτὸν κατέχει µὴ ἐπι πηδᾶν τῷ ἐρωµένῳ). The other one, however, no longer responds to the whip or the goad of the charioteer; it leaps violently forward (σκιρτῶν δὲ βίᾳ φέρεται) and does everything to aggravate its yoke mate and its charioteer, trying to make them go up to the boy and suggest to him the pleasures of sex. At first the other two resist, angry in their belief that they are being made to do things that are dreadfully wrong (τὼ δὲ κατ΄ ἀρχὰς µὲν ἀντιτείνετον ἀγανακτοῦντε͵ ὡς δεινὰ καὶ παράνοµα ἀναγκαζοµένω). At last, however, when they see no end to their trouble, they are led forward, reluctantly agreeing to do as they have been told. (Phdr. 253e5-254b3, trans. Nehamas and Woodruf) What Socrates describes in this passage is very similar to the psychic conflict Leontios endures in book IV of the Republic (439b-c). The verb "ἀγανακτεῖ", used to characterize the entire soul in the previous passage, now refers specifically to the struggle between the charioteer and the good horse on the one hand, and the bad horse on the other. The good horse, thumos, resists by showing self-restraint. The use of the reflexive form "ἑαυτὸν κατέχει" is quite remarkable here, for Plato normally uses these expressions to refer to the way an individual represents himself, torn between several motivations. 28 Here, Socrates seems to refer to the individual not by metonymy, but by synecdoche. In other words, it is as if the good horse were referring to the whole individual, and not one particular part of the soul. Here thumos denotes the real "I", showing self-restraint, representing the emotions as if they were in another part of the soul.

Finally, Socrates gives us a hint in the following description:

At the sight he is frightened (ἔδεισέ), falls over backwards awestruck (σεφθεῖσα), and at the same time has to pull the reins back so fiercely that both horses are set on their haunches, one falling back voluntarily with no resistance, but the other insolent and quite unwilling. They pull back a little further; and while one horse drenches the whole soul with sweat out of shame and awe (ὁ µὲν ὑπ΄ αἰσχύνης τε καὶ θάµβους ἱδρῶτι πᾶσαν ἔβρεξε τὴν ψυχήν), the other -once it has recovered from the pain caused by the bit and its fall -bursts into a torrent of insults (µόγις ἐξαναπνεύσας ἐλοιδόρησεν ὀργῇ) as soon as it has caught its breath, accusing its charioteer and yokemate of all sorts of cowardice and unmanliness for abandoning their position and their agreement (πολλὰ κακίζων τόν τε ἡνίοχον καὶ τὸν ὁµόζυγα ὡς δειλίᾳ τε καὶ ἀνανδρίᾳ λιπόντε τὴν τάξιν καὶ ὁµολογίαν). (Phdr. 254b7-d1, trans. Nehamas and Woodruf).

At first glance, Plato seems to complicate the tripartite model of the soul in ascribing to each part of the soul some features that normally belong to another. The charioteer feels two emotions, fear (deô) and religious awe (sebô), that are traditionally linked with thumos. But, conversely, the bad horse feels anger (orgê), which is, of course, what the thumos feels in Book IV of the Republic (439e-441a). It can be argued that this blurring of symptoms is intentional. All these emotions are, up to a point, one and the same, but seen from different points of view: as in the previous passages, the thumos is made responsible for shame (aischunê); this very same feeling is felt in an intellectualized form by the charioteer (sebas), whereas the bad horse transforms this shame into anger. Thumos, in this description, is the focal point from which stems different perceptions of the beloved, gradually referring to different parts of the soul. To put it as briefly as possible, in these three passages, thumos is both the focal point of the description and the psychic intermediary between reason and appetite. The thumos plays a twofold role: as the first and immediate function of the soul to feel the effects of love, thumos is the main addressee of erotic discourses; but, as an intermediary, it defeats (or tries to defeat) sexual desires, builds a relationship founded on mutual love and respect, and helps give rise to philosophical love. In focusing the description of the effects of love from the viewpoint of thumos, Socrates tries to convert its motivational force into philosophy. Thumoeidic lovers are consequently positively depicted at the end of the myth: If, on the other hand, they adopt a lower way of living, with ambition in place of philosophy (ἀφιλοσόφῳ φιλοτίµῳ δὲ χρήσωνται), then pretty soon when they are careless because they have been drinking or for some other reason, the pair's undisciplined horses will catch their souls off guard (ἀφρούρους) and together bring them to commit that act which ordinary people would take to be the happiest choice of all; and when they have consummated it once, they go on doing this for the rest of their lives, but sparingly, since they have not approved of what they are doing with their whole minds (ἅτε οὐ πάσῃ δεδογµένα τῇ διανοίᾳ πράττοντες). So these two also live in mutual friendship (φίλω µὲν οὖν καὶ τούτω) (though weaker than that of the philosophical pair), both while they are in love and after they have passed beyond it (διά τε τοῦ ἔρωτος καὶ ἔξω γενοµένω διάγουσι), because they realize they have exchanged such firm vows (πίστεις τὰς µεγίστας ἡγουµένω ἀλλήλοιν δεδωκέναι τε καὶ δεδέχθαι) that it would be forbidden (οὐ θεµιτὸν) for them ever to break them and become enemies. In death they are wingless when they leave the body, but their wings are bursting to sprout, so the prize they have won from the madness of love is considerable (οὐ σµικρὸν ἆθλον τῆς ἐρωτικῆς µανίας φέρονται), because those who have begun the sacred journey in lower heaven may not by law be sent into darkness for the journey under the earth; their lives are bright and happy (ἀλλὰ φανὸν βίον διάγοντας εὐδαιµονεῖν) as they travel together, and thanks to their love (ἔρωτος χάριν) they will grow wings together when the time comes. (Phdr. 256b7-e2, trans. Nehamas and Woodruf) Three points must be made on this rehabilitation of thumoeidic lovers. First, their lives are provided with happiness. This happiness is of course less valuable, for it is understood as an intermediary between the regimen of philosophy and what the "ordinary people" would understand as true happiness. Their "friendship" is then weaker than that of philosophers, but nevertheless not to be underestimated (ou smikron). Moreover, this passage should not be understood as a mere hierarchical and static comparison, but as the second prize of a competition; thumoeidic love, made compatible with friendship, should be understood in terms of a mundane and promising temporality. In that sense, philotimia is a necessary intermediary that allows the transformation of a common and popular conception of love into a less bestial relationship. Second, it seems here that an ideal form of erôs is to be found in mutual friendship (philia), challenging the issue of sexual intercourse in the relationship. Sexual intercourse is not absent, but only had seldom, if ever. Again, Socrates seems to stress here the anthropological origin and dynamic of this kind of love. Erôs is not discarded, but modified by the bonds of philia in weakening physical and sexual attachment. Third, philotimia uses thumos' force as a "guard" (phrouros), using social and political bonds: judgments (dogma), marks of trust and confidence (pistis), under the guidance of a rule (themis). These three means are, again, derivative ones compared to reason's rule (and in that respect are "imperfect" ones), but are nevertheless an aid the philosopher uses to shape, as far as possible, human and personal love into a superior type of relationship.

Restrained love in the polis

If after all the Phaedrus does not deny the power of thumos in love, the persuasion and education of the thumos remains a difficult problem. This goal is reached in educating the thumos of individuals and using the very same feelings of shame and sense of honour to regulate erotic relationships in the polis: in the Republic, the Statesman, and the Laws, sexual and marital prescriptions are implied at the very beginning of a broader paideia, to prevent the individual from yielding power to bodily desires and also to a personal attachment that could lead to sexual intercourse. 29 But, more essential to this political program in the Republic and the Laws, is the way Plato transfigures thumos's force in giving it a new object. Traditional feelings such as aidôs and philotimia are dissociated from the object they usually pursue and are used for the purpose of education, for example in the Laws. A famous passage of book VIII in the Laws about the regulation of sexual practices confirms it. As "guardians" of sexual appetites and pleasures (aphrodision), theosebas (a religious awe that inevitably recalls the sebas of the charioteer in the Phaedrus), and philotimia (like Phaedrus' speech in the Symposium) help the individual both to resist sexual attraction and to intellectualize their object of love.

ATHENIAN: My point is that the appetite for pleasures, which is very strong and 29 On the subject, see the thorough analysis of Ludwig (2007), who shows that thumos in the Republic has a "vital connection" to philia and possessiveness (p. 223). I agree with Ludwig that erôs in the ideal state of the Republic should be disconnected from thumos, as far as the philosopher is concerned. But it seems slightly exaggerated to say that "there is a tragic paradox, since the thumos that enables citizen virtue also prevents the ascent to philosophic virtue" (p. 228), for if the citizen in question is definitely a non-philosopher, there is nothing "tragic" about it. But, in the pedagogical and psychagogic context of the Phaedrus, as I have tried to show, even if thumos is inclined to philotimia as such, the object which is valued can be modified.

grows by being fed, can be starved (you remember) if the body is given plenty of hard work to distract it. We'd get much the same result if we were incapable of having sexual intercourse (ἀφροδισίων) without feeling ashamed (ἀναίδεια); our shame (δι΄ αἰσχύνην χρώµενοι) would lead to infrequent indulgence, and infrequent indulgence would make the desire less compulsive (ἀσθενεστέραν ἂν αὐτὴν δέσποιναν). So in sexual matters our citizens ought to regard privacy -though not complete abstinence -as a decency (τι καλὸν) demanded by usage and unwritten custom (νόµιµον ἔθει καὶ ἀγράφῳ νοµισθὲν νόµῳ), and lack of privacy as disgusting (αἰσχρόν). That will establish a second legal standard of decency and indecencynot the standard, but the next to it. People whose characters have been corrupted (they form a single group we call the 'self-inferior') will be made prisoners of three influences that will compel them not to break the law (περιλαβόντα τρία γένη βιάζοιτ΄ ἂν µὴ παρανοµεῖν). CLINIAS: What influences do you mean? ATHENIAN: Respect for religion, the ambition to be honoured, and a mature passion for spiritual rather than physical beauty (Τό τε θεοσεβὲς ἅµα καὶ φιλότιµον καὶ τὸ µὴ τῶν σωµάτων ἀλλὰ τῶν τρόπων τῆς ψυχῆς ὄντων καλῶν γεγονὸς ἐν ἐπιθυµίᾳ). 'Pious wishes!' you'll say; 'what romance!' Perhaps so. But if such wishes were to come true, the world would benefit enormously. (Leg. VIII, 841a6-c8, trans. Saunders)

The Athenian uses shame and philotimia as an ideological way to attain a "second degree of rectitude" for the community's behaviour. For it is not by means of the erotic skills of Socrates that a legislator of a city could lead the citizens to real and philosophical virtue, but instead through the promotion of unwritten law and internalized norms. In this passage, the same feelings which, for example, threatened the philosophical success of Socrates when educating Alcibiades in the Symposium, are used to counteract bodily desires and lust. There is indeed a political and pedagogical use of philotimia in preventing the city from corruption. But this use is ultimately founded on philosophical grounds, according to which thumos, the intermediate part of the soul, can be used as a force for the conversion to philosophy. Plato's task could be summed up as follows: a) given a human nature prone to thumoeidic affections, and b) given a function of the soul that is absolutely central to valuing things, the individual should be educated in a way that c) thumos and generally speaking philotimia should be purified from his concern about timê, and d) thumos should not be an obstacle to seeing the real object of love. Aristotle himself, in book 7 of his Politics, seems to confirm this interpretation, asserting that Plato is well aware of the traditional link between thumos and philia. Recalling how the Guardians in the Republic are organized in a community based on mutual friendship, Aristotle makes thumos the function of the soul whereby we feel philia:

For as to what is said by certain persons about the character that should belong to their Guardians -they should be affectionate to their friends but fierce towards strangers -it is spirit that causes affectionateness (ὁ θυµός ἐστιν ὁ ποιῶν τὸ φιλητικόν), for spirit is the capacity of the soul whereby we love (αὕτη γάρ ἐστιν ἡ τῆς ψυχῆς δύναµις ᾗ φιλοῦµεν). A sign of this is that spirit is more roused against associates and friends than against strangers, when it thinks itself slighted. Therefore, Archilochus for instance, when reproaching his friends, appropriately apostrophizes his spirit: 'For 'tis thy friends that make thee choke with rage' (Pol. VII.7, 1327b38-a5, trans. Rackham).

The expression "the capacity of the soul whereby we love" inevitably recalls the way Socrates introduces the three functions of the soul in book IV of the Republic (436a8-b3). 30 For here, Aristotle plays on two meanings of thumos: as a psychic function, and as pathos akin to anger. The more one feels philia for someone, the more one feels anger in finding oneself betrayed. In a way, thumos as pathos is a negative sign of philia and thus thumos as psychic function can be seen as the seat of philia. However, this passage raises two problems. First, Aristotle does not seem to speak about philia in the broad sense of affectionateness, but in a narrower sense which does not include erôs as physical and sexual attachment. 31 So we might wonder if, in Plato, erôs would essentially be linked with thumos as a function of the soul. Second, what is rather odd in this passage is that Aristotle quotes Archilochus and Euripides as evidence of the traditional link between philia and thumos, not Plato. Actually, Plato did not use this formulation in any dialogue, even in book II of the Republic, which is certainly the passage Aristotle has in mind, in which Socrates says that the thumoeides should not be aggressive towards relatives, balancing its violence with reason's softness (Rsp. 375b-e). But Aristotle seems to take for granted that Plato followed the tradition in placing affectionateness, or more precisely the philia that characterizes the bonds of the Guardians, in thumos. Why so? The reason for such confidence in placing philia in thumos is that the latter is, in Plato's dialogues, the seat of philotimia, a feeling that bonds people (whether citizens or soldiers) together through shared values of mutual assistance and reciprocal esteem. We can now understand why it is important, as Aristotle would claim for philia, to make thumos a central function for love. Even if Plato never said it explicitly, thumos could be the function of the soul whereby we "love", specifically, the seat of the reflexive emotions of shame and the sense of honour which, when they are felt repeatedly in pedagogical training, can defeat a popular and vulgar representation of love.

Conclusion

Platonic love, embodied by a philosophic way of life, must be understood as a model, a new horizon by which other forms of love can be judged and measured. In that respect, philotimia, which may be for us today more akin to "personal love", is certainly criticized as being founded on a wrong image of what deserves to be loved. Nevertheless, philotimia, as the tripartite model of the soul shows, is still an intermediary, more human, maybe less abstract love, through which those who are not philosophers can experience, even imperfectly, some characteristics of philosophic love. Even if the thumoeidic lovers are second in the scale of love, they appear to be a major concern from an anthropological and political perspective. As a psychic guardian, thumos must then be educated to transform the energy of philotimia to restrain physical and sexual desires. Thus, "personal love", even if it is not a valid form of love, remains a starting point before philosophy transforms it.

Olivier RENAUT Université Paris Ouest -Nanterre-La Défense Bibliography 31 I thank C. Gill for having pointed out this issue to me. This chapter of the Politics is entirely devoted to the way the Guardians of the Republic live together. Hence the insistence on the military context of the poetic quotations from Archilochus and Euripides. However, insofar as the community formed by the Guardians in the Republic implies a reform of sexual regulations, we may wonder whether the word "philêtikon" does include more erotic relationships. It is certainly the case in Aristotle's view, but not in Plato's. On this topic, see Leontsini and Gill (this volume).

See the recent attempt of[START_REF] Carone | The Virtues of Platonic Love[END_REF] to combine universal and particular objects of love in the experience of the philosopher in the Symposium.

On erôs and the tyrant, see[START_REF] Larrivée | Malaise dans la cité. Éros et Tyrannie au livre IX de la République[END_REF].

See Nussbaum (2001), ch. 3.

This speech has long been neglected by commentators as a quite conventional speech. However, it makes much more sense to take this passage as a first step towards a transformation of Homeric ethics to fit the Platonic conception of love: seeWersinger (2001) 243-248, and[START_REF] Corrigan | Plato's Dialectic at Play, Argument, Structure and Myth in the Symposium[END_REF] Glazov-Corrigan (2005) 51-56, for an accurate analysis of this speech.

There is some difficulty in translating erôs by love in this passage, for we certainly expect the word philia instead, conveying the idea of a deep attachment rather than a pederastic relationship including sexual attachment. But Phaedrus uses erôs here purposely, as it becomes clear with the examples he chooses at the end of his praise speech: Alcestis, Orpheus and Achilles.

On the meaning of timê, especially in Homeric Epics, see[START_REF] Riedinger | Remarques sur la timê chez Homere[END_REF] who resists the idea that timê is reducible to a mere ransom. For an opposite and resisting view, see[START_REF] Adkins | Values, Goals, and Emotions in the Iliad[END_REF], who seems to acknowledge the traditional platonic criticism of philotimia.

Everything opposes the two discourses. However,[START_REF] Wersinger | Platon et la dysharmonie: Recherches sur la forme musicale[END_REF] 248-257, has shown perfectly well that Phaedrus and Lysias defend an ideal of a relationship based on well-considered charis which implies the self-effacing of the individual. But this reserve (aidôs) does not preclude competitive behaviour forAdkins (1996).

See Phlb. 47e1-48a4, where erôs and thumos are ultimately related to phthonos. See Sanders (this volume), for a jealous woman driven by thumos.

It would be misleading to oppose, as an objection, a so-called "static" hierarchical model of the soul, i.e. the tripartite model with thumos as an intermediary, to the more dynamic and unified one which has been depicted in the Symposium. Whereas[START_REF] Brès | La Psychologie de Platon[END_REF], 308 ff., tries to show that the tripartite model, a static and hierarchical model of the soul, is an impoverishment of a more dynamic psychology which is found in the Symposium,[START_REF] Robin | La Théorie platonicienne de l'amour[END_REF] has consistently shown that this opposition is not relevant, especially in the Phaedrus.

This potential link between the tripartite soul, especially thumos, and philotimia, is clearly presented by[START_REF] Cairns | Aidos: the Psychology and Ethics of Honour and Shame in Ancient Greece[END_REF] 381-9. It might then be tempting to outline analogies between thumos and erôs regarding their function as "intermediaries", as[START_REF] Souilhé | La Notion platonicienne d'intermédiaire dans la philosophie des Dialogues[END_REF] has tried to show, but, despite the resemblance regarding their function, thumos is not necessarily the seat of erôs.

Erôs as sexual appetite has been naturally associated with epithumêtikon in the post-platonic tradition. On this strong association and the physiological interpretation of what is understood as a passion in Galen, see Rosen (this volume).

See for example Rsp. 387b8-c5. Especially, the verbs "ζεῖν" and "ἀγανακτεῖν" Socrates uses to refer to thumoeidic character, or more precisely to thumos itself: Cra. 419e, Phd. 64a8, Rsp. 440c7 and 536c1, and Ti. 70b3.

See for example Thrasymachus in Rsp. 350c12-d3. See also Ti. 84d2-e2. On the poetic legacy of the imagery of the charioteer and his symptoms in the Phaedrus, see Cairns (this volume).

See for example Rsp. 430e-431a, in which Socrates analyses the meaning of the expression "to be master of oneself".

See also. For a clear discussion on the use of this formulation (instrumental dative + verb), seeMacé (2006), 89-90, 189.


Gabriel Peyré

Gabriel Peyre

Marco Cuturi
email: mcuturi@i.kyoto-

Justin Solomon
email: jsolomon@mit.edu

Gromov-Wasserstein Averaging of Kernel and Distance Matrices

This paper presents a new technique for computing the barycenter of a set of distance or kernel matrices. These matrices, which define the interrelationships between points sampled from individual domains, are not required to have the same size or to be in row-by-row correspondence. We compare these matrices using the softassign criterion, which measures the minimum distortion induced by a probabilistic map from the rows of one similarity matrix to the rows of another; this criterion amounts to a regularized version of the Gromov-Wasserstein (GW) distance between metric-measure spaces. The barycenter is then defined as a Fréchet mean of the input matrices with respect to this criterion, minimizing a weighted sum of softassign values. We provide a fast iterative algorithm for the resulting nonconvex optimization problem, built upon state-ofthe-art tools for regularized optimal transportation. We demonstrate its application to the computation of shape barycenters and to the prediction of energy levels from molecular configurations in quantum chemistry.

Introduction

Many classes of input data encountered in machine learning are best expressed using either pairwise distance matrices or kernels. For instance, the atoms making up proteins and other molecules have no canonical orientation and hence are only known up to rigid motion; so, the geometry of a molecule might be better described by the pair- wise distances between atoms rather than by their absolute positions. Kernel and covariance matrices from different datasets exhibit similar structure, since they define data points by their roles relative to the rest of a dataset rather than in any absolute sense.

A major difficulty with this representation is that in many applications these matrices are not "registered" or "aligned," meaning that there is no explicit correspondence between their rows and columns. This is the case for shapes that undergo pose variation, articulation or deformation. Even worse, in some cases, similarities are not defined over the same ground space. For example, different molecules may have varying numbers of atoms. This inconsistency yields matrices of varying dimensions.

In this paper, we propose a theoretical and computational framework for summarizing collections of unaligned distance or kernel matrices. Building upon the notion of Gromov-Wasserstein (GW) distances between metricmeasure spaces [START_REF] Mémoli | On the use of Gromov-Hausdorff distances for shape comparison[END_REF], we provide machinery for registering matrices of varying sizes while building interpolants and barycenters inheriting structure from the inputs. We also derive a fast regularized approximation scheme providing a practical and efficient way to recover these barycenters in practice.

Previous Work

Countless techniques leverage matrices of distances or kernels to describe a dataset. While distances and kernels have opposite ordering-a large distance indicates small similarity-unless it is necessary to distinguish we will use the term "similarity matrix" to refer to any matrix containing pairwise relationships.

Aligning similarity matrices. Comparison of similarity matrices is challenging due to a lack of alignment. Mathematically, they only define the intrinsic structure of a dataset, which remains unchanged e.g. under rotation, translation, and isometric motion. More concretely, there is unlikely to be a canonical ordering of the rows or columns of similarity matrices, and removing or adding a few rows/columns may not significantly affect the structure. Automatic computation of a map aligning two similarity matrices is itself a challenging problem; our work is fairly unique in that we simultaneously compute such an alignment while solving an optimization problem over such matrices.

Matching rows of similarity matrices usually becomes a quadratic assignment problem [START_REF] Loiola | A survey for the quadratic assignment problem[END_REF], which is NP-hard. For the specific case of aligning similarity matrices derived from pairwise distances in a metric space (e.g., geodesic distances along a surface), a well-known instance of this matching problem is computation of the Gromov-Hausdorff distance [START_REF] Gromov | Metric Structures for Riemannian and Non-Riemannian Spaces[END_REF], which was applied to matching in [START_REF] Mémoli | On the use of Gromov-Hausdorff distances for shape comparison[END_REF].

To relax the matching problem to a continuous-but still non-convex-optimization problem, the seminal work of Mémoli considers matching between metric-measure spaces, incorporating a probability distribution on the ground space that can account for some form of uncertainty [START_REF] Mémoli | Gromov-Wasserstein distances and the metric approach to object matching[END_REF]. The geodesic properties of this "space of metric spaces" are considered in [START_REF] Sturm | The space of spaces: curvature bounds and gradient flows on the space of metric measure spaces[END_REF]. The recent work [START_REF] Hendrikson | Using Gromov-Wasserstein distance to explore sets of networks[END_REF] showcases the application of the corresponding Gromov-Wasserstein distance to clustering of biological and social networks. It is possible to further relax this formulation to a quadratic [START_REF] Aflalo | On convex relaxation of graph isomorphism[END_REF] or semidefinite [START_REF] Kezurer | Tight relaxation of quadratic matching[END_REF] convex program, although this relaxation can fail if the input similarity matrices exhibit symmetries.

Rather than mixing the possible symmetric maps, we embrace the nonconvexity of the original problem and optimize quadratic matching objectives directly. After regularization, our matching subroutine resembles "softassign quadratic assignment" [START_REF] Rangarajan | Convergence properties of the softassign quadratic assignment algorithm[END_REF][START_REF] Gold | A graduated assignment algorithm for graph matching[END_REF]. We extend their iterations to general loss functions and to more advanced machine learning problems than nearest-neighbor search, in particular barycenter computation. We additionally leverage the fact that GW can compare arbitrary matrices, not just pairwise distances, making our machinery applicable to a broader range of tasks. The mapping component of our algorithm can be considered a generalized version of the method recently proposed in [START_REF] Solomon | Entropic metric alignment for correspondence problems[END_REF].

Averaging similarity matrices. The design of an "average" or barycenter of similarity matrices is informed by two primary factors: A measure of discrepancy between similarity matrices and a definition of a mean. We will GW distances to define discrepancies, so our remaining task is to define what it means to compute a barycenter.

Even when similarity matrices are the same size and registered, it may not make sense to average them arithmetically. One natural way to define a mean or barycenter of inputs (C s ) s is to minimize the sum s λ s d 2 (C, C s ) over all possible C, where d is some notion of distance. This generalized notion of a mean is known as the Karcher or Fréchet mean, which generalizes the notion of an average to a wide class of metric spaces [START_REF] Nielsen | Matrix Information Geometry[END_REF].

Fréchet barycenters in Wasserstein space were proposed in [START_REF] Agueh | Barycenters in the Wasserstein space[END_REF]. Several algorithms for their computation have been proposed including [START_REF] Cuturi | Fast computation of Wasserstein barycenters[END_REF][START_REF] Benamou | Iterative Bregman projections for regularized transportation problems[END_REF]; these assume a fixed ground distance metric. Our method, however, finds more commonality with algorithms that compute Fréchet means in the space of covariance or metric matrices. [START_REF] Dryden | Non-Euclidean statistics for covariance matrices, with applications to diffusion tensor imaging[END_REF] considers several simple options for covariance matrices, mostly with fixed alignment; their "Procrustes" mean provides one rudimentary strategy for alignment. [START_REF] Sra | Positive definite matrices and the S-divergence[END_REF] defines a metric on registered positive definite kernel matrices that can be used for computing Fréchet means.

Barycenter computation is a building block for many learning methods. For instance, k-means clustering alternates between distances and barycenter computation. The nearest-centroid classifier provides a similar approach to classification [START_REF] Manning | Introduction to Information Retrieval[END_REF].

Optimal transport (OT). OT [START_REF] Villani | Topics in Optimal Transportation[END_REF] is a way to compare probability distributions (histograms in the finitedimensional case) defined over either the same ground space or multiple pre-registered ground spaces. The means of comparison is a convex linear program optimizing for a matching that moves the mass from one distribution to the other with minimal cost. OT has been applied to machine learning problems including comparison of descriptors [START_REF] Cuturi | Sinkhorn distances: Lightspeed computation of optimal transportation[END_REF], k-means [START_REF] Cuturi | Fast computation of Wasserstein barycenters[END_REF], semi-supervised learning [START_REF] Solomon | Earth mover's distances on discrete surfaces[END_REF], domain adaptation [START_REF] Courty | Domain adaptation with regularized optimal transport[END_REF], designing loss functions [START_REF] Frogner | Learning with a Wasserstein loss[END_REF], low-rank approximation [START_REF] Seguy | Principal geodesic analysis for probability measures under the optimal transport metric[END_REF], and dictionary learning [START_REF] Rolet | Fast dictionary learning with a smoothed Wasserstein loss[END_REF].

As highlighted above, GW-style matching extends OT to the case when the ground spaces are not pre-registered, yielding a non-convex quadratic program to compute the transport. The resulting transportation matrix can be understood is a soft registration from one domain to the other. Our algorithm solves a entropically-regularized version of this quadratic program by extending recent advances in the computation of OT [START_REF] Cuturi | Sinkhorn distances: Lightspeed computation of optimal transportation[END_REF][START_REF] Benamou | Iterative Bregman projections for regularized transportation problems[END_REF].

Contributions

Our first contribution is the definition of a new discrepancy between similarity matrices. It extends the "Gromov-Wasserstein" distance between metric-measure spaces to arbitrary matrices, using a generic loss functions to compare pairwise similarities and entropic regularization. It can be defined over different ground measured spaces (i.e. each point is equipped with a measure), which are not required to be registered a priori. Entropic regularization enables the design of a fast iterative algorithm to compute a stationary point of the non-convex energy defining the discrepancy.

Our second contribution is a new notion of the barycenter of a set of unregistered similarity matrices, defined as a Fréchet mean with respect to the GW discrepancy. We propose a block coordinate relaxation algorithm to compute a stationary point of the objective function defining our barycenter.

We showcase applications of our method to the computation of barycenters between shapes. We also exemplify how the GW discrepancy can be used to predict energy levels in quantum chemistry, where molecules are naturally represented using their Coulomb interaction matrices, a perfect fit for our unregistered dissimilarity matrix formalism.

The code to reproduce the results of this paper is available online.1

Notation

The simplex of histograms with N bins is

Σ N def. = p ∈ R + N ; i p i = 1 . The entropy of T ∈ R N ×N + is defined as H(T ) def. = - N i,j=1 T i,j (log(T i,j ) -1). The set of couplings between histograms p ∈ Σ N1 and q ∈ Σ N2 is C p,q def. = T ∈ (R + ) N1×N2 ; T 1 N2 = p, T 1 N1 = q .
Here, 1 N def.

= (1, . . . , 1) ∈ R N . For any tensor L = (L i,j,k, ) i,j,k, and matrix (T i,j ) i,j , we define the tensormatrix multiplication as

L ⊗ T def. = k, L i,j,k, T k, i,j
.

(1)

Gromov-Wasserstein Discrepancy

Entropic Optimal Transport

Optimal transport distances are useful to compare two histograms (p, q) ∈ Σ N1 × Σ N2 defined on the same metric space, or at least on spaces that have previously registered. Given some cost matrix c ∈ R N1×N2

+

, where c i,j represents the transportation cost between position indexed by i and j, we define the solution of entropically-regularized optimal transport between these two histograms as T (c, p, q) def.

= argmin

T ∈Cp,q c, T -εH(T ), (2)
which is a strictly convex optimization problem.

As shown in [START_REF] Cuturi | Sinkhorn distances: Lightspeed computation of optimal transportation[END_REF], the solution reads

T (c, p, q) = diag(a)K diag(b) where K def. = e -c ε ∈ R N1×N2 + is the so- called Gibbs kernel associated to c, and (a, b) ∈ R N1 + ×R N2 + can be computed using Sinkhorn iterations a ← p Kb and b ← q K a , (3)
where here • • denotes component-wise division.

Gromov-Wasserstein Discrepancy

Following the pioneering work of Mémoli, we consider input data expressed as metric-measure spaces [START_REF] Mémoli | Gromov-Wasserstein distances and the metric approach to object matching[END_REF]. This corresponds to pairs of the form (C, p) ∈ R N ×N × Σ N , where N is an arbitrary integer (the number of elements in the underlying space). Here, C is a matrix representing either similarities or distances between these elements, and p is an histogram, which can account either for some uncertainty or relative importance between these elements. In case no prior information is known about a space, one can set p = 1 N 1 N to the uniform distribution. In our setting, since we target a wide range of machinelearning problems, we do not restrict the matrices C to be distance matrices, i.e., they are not necessarily positive and does not necessarily satisfy the triangle inequality.

We define the Gromov-Wasserstein discrepancy between two measured similarity matrices (C, p) ∈ R N1×N1 × Σ N1 and ( C, q) ∈ R N2×N2 × Σ N2 as follows:

GW(C, C, p, q) def. = min T ∈Cp,q E C, C (T ) (4)
where

E C, C (T ) def. = i,j,k, L(C i,k , Cj, )T i,j T k,
The matrix T is a coupling between the two spaces on which the similarity matrices C and C are defined. Here L is some loss function to account for the misfit between the similarity matrices. Typical choices of loss include the quadratic loss

L(a, b) = L 2 (a, b) def.
= 1 2 |a -b| 2 and the Kullback-Leibler divergence L(a, b) = KL(a|b) def.

= a log(a/b) -a + b (which is not symmetric). This definition (4) of GW extends slightly the one considered by [START_REF] Mémoli | Gromov-Wasserstein distances and the metric approach to object matching[END_REF], since we consider an arbitrary loss L (rather than just the L 2 squared loss). In the case [START_REF] Mémoli | Gromov-Wasserstein distances and the metric approach to object matching[END_REF] proves that GW 1/2 defines a distance on the space of metric measure spaces quotiented by measure-preserving isometries.

L = L 2 ,
Introducing the 4-way tensor

L(C, C) def. = (L(C i,k , Cj, )) i,j,k, , we notice that E C, C (T ) = L(C, C) ⊗ T, T ,
where ⊗ denotes the tensor-matrix multiplication (1).

The following proposition shows how to compute L(C, C) ⊗ T efficiently for a general class of loss functions:

Proposition 1. If the loss can be written as

L(a, b) = f 1 (a) + f 2 (b) -h 1 (a)h 2 (b) (5) for functions (f 1 , f 2 , h 1 , h 2 ), then, for any T ∈ C p,q , L(C, C) ⊗ T = c C, C -h 1 (C)T h 2 ( C) . ( 6
)
where c C,

C def. = f 1 (C)p1 N2 +1 N1 q f 2 ( C) is independent of T .
Proof. Under hypothesis (5), formula (1) shows that one has the decomposition L(C, C) ⊗ T = A + B + C where

A i,j = k f 1 (C i,k ) T k, = (f 1 (C)(T 1)) i B i,j = f 2 ( Cj, ) k T k, = (f 2 ( C)(T 1)) j C i,j = k h 1 (C i,k ) h 2 ( Cj, )T k, which is equal to (h 1 (C)(h 1 ( C)T ) ) i,j .
Remark 1 (Computational complexity). Formula (6) shows that for this class of losses, one can compute

L(C, C) ⊗ T efficiently in O(N 2 1 N 2 +N 2 2 N 1 ) operations, using only ma- trix/matrix multiplications, instead of the O(N 2 1 N 2 2 ) com- plexity of the naïve implementation of formula (1). Remark 2 (Special cases). The square loss L = L 2 sat- isfies (5) for f 1 (a) = a 2 , f 2 (b) = b 2 , h 1 (a) = a and h 2 (b) = 2b. The KL loss L = KL satisfies (5) for f 1 (a) = a log(a) -a, f 2 (b) = b, h 1 (a) = a and h 2 (b) = log(b).

Entropic Gromov-Wasserstein Discrepancy

We consider the following entropic approximation of the initial GW formulation (4)

GW ε (C, C, p, q) def. = min T ∈Cp,q E C, C (T ) -εH(T ). (7)
This is a non-convex optimization problem. We propose to use projected gradient descent, where both the gradient step and the projection are computed according to the KL metric.

Iterations of this algorithm are given by

T ← Proj KL Cp,q T e -τ (∇E C, C (T )-ε∇H(T )) , (8)
where τ > 0 is a small enough step size, and the KL projector of any matrix K is

Proj KL Cp,q (K)
def.

= argmin

T ∈Cp,q KL(T |K).

Proposition 2. In the special case τ = 1/ε, iteration (8

) reads T ← T (L(C, C) ⊗ T, p, q). ( 9
)
Proof. As shown in [START_REF] Benamou | Iterative Bregman projections for regularized transportation problems[END_REF], the projection is nothing else than the solution to the regularized transport problem (2), hence

Proj KL Cp,q (K) = T (-ε log(K), p, q).
One also has

∇E C, C (T ) -ε∇H(T ) = L(C, C) ⊗ T + ε log(T ).
Re-arranging the terms in (8), one obtains, in the special case τ ε = 1, the desired formula.

Iteration (9) defines a surprisingly simple algorithm, in which each update of T involves a Sinkhorn projection.

Remark 3 (Convergence). Using [START_REF] Bot | An inertial forward-backward algorithm for the minimization of the sum of two nonconvex functions[END_REF] (see Theorem 12), iterations (8) are guaranteed to converge provided that τ is chosen small enough, τ < τ max . Unfortunately, in general, one does not have τ max > 1/ε, so that the step size τ = 1/ε advocated by Proposition 2 is not covered by the theory. However, we found that using τ = 1/ε always leads to a converging sequence of T , and that this works well in practice. Note that when L = L 2 , this recovers the "softassign quadratic assignement" algorithms defined in [START_REF] Rangarajan | Convergence properties of the softassign quadratic assignment algorithm[END_REF][START_REF] Gold | A graduated assignment algorithm for graph matching[END_REF]. The convergence proof [START_REF] Rangarajan | Convergence properties of the softassign quadratic assignment algorithm[END_REF], however, only ensures convergence of the functional values (not of the iterates) and also only applies when the function being minimized in (7) is convex, which is not the case for arbitrary matrices (C, C s ).

Gromov-Wasserstein Barycenters

Gromov-Wasserstein Barycenters

We define Gromov-Wasserstein barycenters of measured similarity matrices (C s ) S s=1 , where C s ∈ R Ns×Ns , using a Fréchet mean formulation:

min C∈R N ×N s λ s GW ε (C, C s , p, p s ). (10)
In this section, for simplicity, we assume that the base histograms (p s ) s and the histogram p associated to the barycenter are known and fixed. Note in particular that the size (N, N ) of the targeted barycenter matrix should be fixed by the user. It is straightforward to extend the exposition as well as the optimization algorithm to the setting where p is unknown and included as an optimization variable.

This barycenter can be reformulated by re-introducing couplings min C,(Ts)s s

λ s (E C,Cs (T s ) -εH(T s )) (11)
subject to the constraints that for all s, T s ∈ C p,ps ⊂ R N ×Ns + .

Note that if L is convex with respect to its first variable, this problem is convex with respect to C but not with respect to (T s ) s (it is a quadratic but not necessarily positive problem with respect to the (T s ) s ).

We propose to minimize (11) using a block coordinate relaxation, i.e. iteratively minimizing with respect to the couplings (T s ) s and to the metric C.

Minimization with respect to (T s ) s . The optimization problem (10) over (T s ) s alone decouples as S independent GW ε optimizations ∀ s, min

Ts∈Cp,p s E C,Cs (T s ) -εH(T s ).
A stationary point of this optimization problem can be computed using the optimization algorithm detailed in Section 2.3.

Minimization with respect to C. For given (T s ), the minimization with respect to C reads

min C s λ s L(C, C s ) ⊗ T, T . (12)
The following proposition shows, for a large class of losses, how to compute the global minimizer in closed form.

Proposition 3. If L satisfies (5) and f 1 /h 1 is invertible, then the solution to (12) reads

C = f 1 h 1 -1 s λ s T s h 2 (C s )T s pp , (13)
where we assume the normalization s λ s = 1.

Proof. Using relation ( 6), the functional to be minimized reads

s λ s f 1 (C)p1 + 1p s f 2 (C s ) -h 1 (C)T s h 2 (C s ) , T s .
The first order optimality condition for this optimization problem thus reads

f 1 (C) (pp ) = h 1 (C)
s λ s T s h 2 (C s )T s , which gives the desired formula.

The intuition underlying formula ( 13) is clear: Each T s h 2 (C s )T s is a "realigned" matrix where T s acts as a fuzzy permutation (optimal transportation coupling) of both rows and columns of the distance matrix C s . These realigned metric are then averaged, where the precise notion of "averaging" depends on the loss L.

Barycenters of PSD kernels using L 2 loss. For the square loss L = L 2 , the update (13) becomes

C ← 1 pp s λ s T s C s T s . (14)
This formula highlights an important property of the method:

Proposition 4. For L = L 2 , if the (C s ) s are positive semidefinite (PSD) matrices, the iterates C produced by the algorithm are also PSD.

Proof. Formula (14) shows that the update of C corresponds to a linear averaging of the matrices (diag(1/p)T s C s T s diag(1/p)) s , which are all PSD since the matrices (C s ) s are.

This proposition implies, since the SDP cone is a closed set, that the output of our barycenter algorithm at convergence is an SDP kernel.

Barycenters of infinitely divisible kernels using KL loss.

For the KL loss L = KL, the update (13) similar, but with a geometric mean in place of an arithmetic mean

C ← exp 1 pp s λ s T s log(C s )T s . ( 15
)
The following proposition shows that this loss is particularly attractive for averaging infinitely divisible kernels.

Recall that C ∈ R N ×N is infinitely divisible if and only if U = log(C) is a conditionally positive semidefinite kernel, i.e. for all x ∈ R N such that x, 1 N = 0, then U x, x 0.

Proposition 5. For L = KL, if the (C s ) s are infinitely divisible kernels, the iterates C produced by the algorithm are also infinitely divisible. // Sinkhorn iterations (3) to compute T (c s , p, q)

Initialize a ← 1, set K ← e -cs/ε . repeat a ← p Kb , b ← q K a . until convergence Update T s ← diag(a)K diag(b).
until convergence end for // minimize over C using (13).

C ← f 1 h 1 -1 s λsT s h2(Cs)Ts pp until convergence
Proof. According to formula ( 15), one has to show that U def.

= s λ s diag(1/p)T s U s T s diag(1/p) is conditionally PSD prodived that the U s matrices are. This is indeed the case, because, for any x such that x, 1 N = 0, one has U x, x = s λ s U s x s , x s where x s def.

= T s x p , and since x s , 1 Ns = x p , T s 1 Ns = x p , p = 0, one has U s x s , x s 0 so that U x, x 0, which proves the result.

This proposition implies, since the cone of infinitely divisible kernels is a closed set, that the output of our barycenter algorithm at convergence is infinitely divisible.

A important example of infinitely divisible kernels is given by the form C = e -D 2 σ 2 , where D i,j = ||x i -x j || is the Euclidean distance matrix between points (x i ) i , and σ > 0 is a bandwidth parameter that controls the "locality" of the resulting kernel. One verifies that KL(e

-D 2 σ 2 |e -D2 σ 2 ) = 1 2σ 4 ||D 2 -D2 || 2 + o(1/σ 4
), so that in the regime of large bandwidth σ → +∞, using the KL loss in conjunction with this class of kernels becomes equivalent to using the L 2 loss between squared Euclidean matrices. A similar result in fact holds for any smooth Bregman divergence loss.

Pseudocode. Algorithm 1 details the steps of the optimization technique. Note that it makes use of three nested iterations: (i) blockwise coordinate descent on (T s ) s and C, (ii) projected gradient descent on each T s , and (iii) Sinkhorn iterations to compute the projection. . Barycenter example for shape data from [START_REF] Thakoor | Hidden Markov model-based weighted likelihood discriminant for 2-D shape classification[END_REF].

Experiments

Point Clouds

Embedded barycenters. Figure 1 provides an example illustrating the behavior of our GW barycenter approximation. In this experiment, we extract 500 point clouds of handwritten digits from the dataset [START_REF] Lecun | Gradient-based learning applied to document recognition[END_REF], rotated arbitrarily in the plane. We represent each digit as a symmetric Euclidean distance matrix and optimize for a 500 × 500 barycenter using Algorithm 1 (uniform weights, ε = 1 × 10 -3 ); notice that most of the input point clouds consist of fewer than 500 points. We then visualize the barycenter matrix as a point cloud in the plane using multidimensional scaling (MDS). Each digit is colored by transferring RGB values from the barycenter point cloud using the computed map T .

This experiment illustrates a few properties of our algorithm. Most prominently, note that the input digits are rotated arbitrarily, which-unlike tools based on classical optimal transportation-does not affect the computation of the barycenter. Also, to avoid bias in this experiment we initialize C as a random matrix, and yet our nonconvex optimization algorithm still reaches a meaningful barycenter.

Figure 2 illustrates a similar experiment on the 2D shape data from [START_REF] Thakoor | Hidden Markov model-based weighted likelihood discriminant for 2-D shape classification[END_REF]. This second experiment illustrates some additional properties of our barycenters. Interestingly, even though we do not impose curve topol-

ε = 1.5 × 10 -3 ε = 4 × 10 -3 ε = 8 × 10 -3 ε = 1.6 × 10 -2 ε = 3.2 × 10 -2
It shows the barycenter from a test in Figure 2 for increasing values of ε. When ε is small, the embedded barycenter most closely resembles the input data. As ε increases, the barycenters become smoother, with the advantage of higher resilience to noise and fewer iterations needed for optimization. In this example, we construct a dataset with 50 point clouds from each of digits 0 to 4 from the example in Figure 1. We then cluster the point clouds by representing them as pairwise distance matrices and applying the k-means algorithm (k = 5), with k-means++ initialization [START_REF] Arthur | K-means++: The advantages of careful seeding[END_REF]. We underscore that each handwritten character is represented using an unaligned point cloud with varying numbers of points.

Clustering

Each row in the figure corresponds to a different character, and each column corresponds to a different cluster center (MDS embedding shown above the table). Color is proportional to the number of data points assigned to each cluster. Even without supervision, this rudimentary technique for clustering finds meaningful clusters of handwritten digits from the dataset. All the "1" digits are clustered cor- = ||x s,i -x s,j || for s ∈ {1, 2} and i, j = 1, . . . , N . The interpolation is achieved by first computing a barycenter matrix C t minimizing (10) with our optimization algorithm, for λ = (t, 1 -t), t ∈ [0, 1], using the Euclidean loss L = L 2 . A barycentric point cloud xt = (x t,i ) N i=1 is then reconstructed using SMACOF multidimensional scaling (MDS), which amounts to computing a local minimizer of i,j |C t,i,j -||x t,i -xt,j ||| 2 with respect to (x t,i ) i . This local minimizer xt is computed by a gradient descent scheme, initialized with x 1 .

L 2 vs. KL losses. Euclidean vs. Geodesic distance matrices. Euclidean pairwise distance, though simple to compute and to manipulate, fail to capture the intrinsic geometry of shapes, and thus leads to physically unrealistic shape interpolation, that ignore in particular articulation. Figure 4, lower part, shows how this issue can be in large part alleviated by replacing the extrinsic Euclidean distance C s,i,j = ||x s,ix s,j || for s ∈ {1, 2} by the inner geodesic distance (i.e. the length of the shortest path joining x s,i to x s,j while staying inside the shape). Note that, because the corresponding distance is not Euclidean anymore, the SMACOF MDS only produces an approximate embedding, the so-called bending invariant [START_REF] Elad | On bending invariant signatures for surfaces[END_REF] used to perform isomericinvariant shape recognition.

Quantum chemistry

To demonstrate the interest of the Gromov-Wasserstein discrepancy, we consider its application to a regression problem on molecules. Several recent works [START_REF] Rupp | Fast and accurate modeling of molecular atomization energies with machine learning[END_REF][START_REF] Hansen | Assessment and validation of machine learning methods for predicting molecular atomization energies[END_REF] have proposed to predict atomization energies for molecules using descriptors of labeled molecules, rather than estimating them through expensive numerical simulations. [START_REF] Rupp | Fast and accurate modeling of molecular atomization energies with machine learning[END_REF] proposed in particular to represent each molecule in the qm7 dataset of organic 7165 molecules through its so-called "Coulomb matrix," which we describe next.

Coulomb matrices. For a molecule s with N s atoms, each described by a relative location r i ∈ R 3 in space and a nuclear charge Z i , Rupp et al. proposed to form the N s × N s matrix C s with off-diagonal terms Z i Z j /||r i -r j || for 1 i = j N s , and diagonal terms 1 2 Z 2.4 i . Although this plays no major role in our analysis, we found that these 7165 Coulomb matrices are all infinitely divisible positive definite kernel matrices. Rupp et al. argue that the Coulomb matrix is a convenient descriptor for molecules in the sense that it is invariant with respect to translations and rotations by construction. It has, however, an important weakness: C s has an arbitrary ordering of the N s atoms in s. As we explain next, this was addressed by Rupp et al. by creating randomly permuted copies of these matrices.

Previous work. Because their approach requires manipulating matrices of the same size, Rupp et al. pad all N s ×N s Coulomb matrices C s with zeros to obtain 23×23 matrices (23 is the maximal number of atoms in a molecule in the qm7 database). To cope with the ordering problem, they also propose, as a representation for each molecule, to form several randomly permuted copies of C s for each molecule s (between 8 and 1000 in [START_REF] Hansen | Assessment and validation of machine learning methods for predicting molecular atomization energies[END_REF]). (Hansen et al., 2013, Table 3) provide out-of-sample mean absolute error (MAE) predictions for several techniques using 5 fold cross-validation. We report some of them in [START_REF] Hansen | Assessment and validation of machine learning methods for predicting molecular atomization energies[END_REF], Table 3) except ours. Number in between parenthesis stand for random copies used in the algorithm. Although the GW distance is not directly competitive performance-wise, this result shows that an acceptable performance on this task can be recovered exclusively using simple metric tools.

Our approach. We propose to use the original N s × N s Coulomb matrices C s directly as inputs for our GW distances. Namely, we compute a discrepancy between two molecules s, s that is exactly GW(C s , C s , 1 Ns /N s , 1 N s /N s ), using L = L 2 and a regularization strength ε tuned heuristically on a subset of 500 points of the database. Using a 3-nearest neighbor regression approach we obtain a MAE of 10.83 which, although not directly competitive with the best approach recorded in [START_REF] Hansen | Assessment and validation of machine learning methods for predicting molecular atomization energies[END_REF], it remarkable in the sense that it is obtained with an extremely simple classifier. Our result contradicts thus the observation made by Hansen et al.,p.3414,that This [poor performance of k-nn] indicates that there are meaningful linear relations between physical quantities in the system and that it is insufficient to simply look up the most similar molecules (as k-nearest neighbors does). The k-nearest neighbors approach fails to create a smooth mapping to the energies.

Conclusion

The Gromov-Wasserstein discrepancy measure is relevant to countless problems in machine learning for which the primary role of a given data point is defined relative to other members of the dataset. After introducing an entropic regularizer to the problem, we in particular find that GW is not only theoretically attractive but also a practical tool with an efficient and easily-implemented non-convex optimization scheme. Our approach highlights the fact that using unregistered similarity matrices as features in machine learning problems is fruitful in many contexts. GW provides a natural framework for manipulation of these matrices, as we demonstrate on classification, clustering, and regression tasks for shapes and molecules. We suspect this elegant and efficient solution to extend to many other problem instances endowed with natural geometric structure.

Figure 1 .

 1 Figure 1. Barycenters of point clouds from MNIST digits; sample input data clouds are shown on the left, and an MDS embedding of the barycenter distance matrix is shown on the right.

Figure 2

 2 Figure2. Barycenter example for shape data from[START_REF] Thakoor | Hidden Markov model-based weighted likelihood discriminant for 2-D shape classification[END_REF].

Figure 3 .

 3 Figure 3. Progressive interpolation (xt) 1 t=0 between two shapes (x0, x1) using GW barycenters.

.

  The figure on the right illustrates the incorporation of our barycenter technique into a larger machine learning pipeline. It shows results of a k-means unsupervised clustering of 250 handwritten characters.

Figure 4 .

 4 Figure 4. Upper part: comparison between interpolation using L 2 loss and KL loss. Lower part: comparison between interpolation using pairwise Euclidean and inner geodesic distances.

  Figure 4, upper part, shows a comparison of the same interpolation as those computed in Figure 3, computed with two different losses. Row #1 shows barycenters of the distance matrices C s,i,j = ||x s,i -x s,j ||, for s ∈ {1, 2}, using the squared Euclidean loss L = L 2 as in the previous paragraph. The display is obtained by applying SMACOF MDS to the barycenter matrix C t . Row #2 shows barycenter of the infinitely divisible kernels C s,i,j = exp(-||x s,i -x s,j || 2 /σ 2 ), for s = {1, 2}, using the Kullback-Leibler loss L =KL, and a bandwidth σ = 1 (for point clouds in the unit square). The display is then obtained by applying SMACOF MDS to -σ log(C t ). The KL interpolation produces less regular interpolation because it imposes only local constraints at the scale of the bandwidth.

  Proceedings of the 33 rd International Conference on Machine Learning, New York, NY, USA, 2016. JMLR: W&CP volume 48. Copyright 2016 by the author(s).

  Algorithm 1 Computation of GW ε barycenters. Input: (C s , p s ) s , p Initialize C. repeat // minimize over (T s ) s for s = 1 to S do Initialize T s . repeat // compute c s = L(C, C s ) ⊗ T s using (6).

	cs ← f1(C)+ f2(Cs) -h1(C)Tsh2(Cs)

Table 1 .

 1 Table 4.2. Mean-Absolute and Root Mean Squared errors for the atomization energy prediction in the qm7 database of 7165 molecules. All results quoted from

	Algorithm	MAE RMSE
	k-nearest neighbors	71.54 95.97
	Linear regression	20.72 27.22
	Gaussian kernel ridge regression	8.57	12.26
	Laplacian kernel ridge regression (8) 3.07	4.84
	Multilayer Neural Network (1000)	3.51	5.96
	GW 3-nearest neighbors	10.83 29.27

https://github.com/gpeyre/

2016-ICML-gromov-wasserstein

Acknowledgements The work of G. Peyré has been supported by the European Research Council (ERC project SIGMA-Vision). M. Cuturi gratefully acknowledges the support of JSPS young research A grant 26700002. J. Solomon acknowledges the support of the NSF Mathematical Sciences Postdoctoral Research Fellowship (award number 1502435).