arXiv subjects
Publications and source records attributed to Frank Nielsen.
This letter introduces an abstract learning problem called the "set embedding": The objective is to map sets into probability distributions so as to lose less information. We relate set union and intersection operations with corresponding interpolations of probability distributions. We also demonstrate a preliminary solution with experimental results on toy set embedding examples.
We first introduce the class of strictly quasiconvex and strictly quasiconcave Jensen divergences which are oriented (asymmetric) distances, and study some of their properties. We then define the strictly quasiconvex Bregman divergences as the limit case of scaled and skewed quasiconvex Jensen divergences, and report a simple closed-form formula which shows that these divergences are only pseudo-divergences at countably many inflection points of the generators. To remedy this problem, we propose the $δ$-averaged quasiconvex Bregman divergences which integrate the pseudo-divergences over a small neighborhood in order obtain a proper divergence. The formula of $δ$-averaged quasiconvex Bregman divergences extend even to non-differentiable strictly quasiconvex generators. These quasiconvex Bregman divergences between distinct elements have the property to always have one orientation finite while the other orientation is infinite. We show that these quasiconvex Bregman divergences can also be interpreted as limit cases of generalized skewed Jensen divergences with respect to comparative convexity by using power means. Finally, we illustrate how these quasiconvex Bregman divergences naturally appear as equivalent divergences for the Kullback-Leibler divergences between probability densities belonging to a same parametric family of distributions with nested supports.
This paper presents the Variation Network (VarNet), a generative model providing means to manipulate the high-level attributes of a given input. The originality of our approach is that VarNet is not only capable of handling pre-defined attributes but can also learn the relevant attributes of the dataset by itself. These two settings can also be easily considered at the same time, which makes this model applicable to a wide variety of tasks. Further, VarNet has a sound information-theoretic interpretation which grants us with interpretable means to control how these high-level attributes are learned. We demonstrate experimentally that this model is capable of performing interesting input manipulation and that the learned attributes are relevant and meaningful.
This paper is motivated by real-life applications of bi-objective optimization. Having many non dominated solutions, one wishes to cluster the Pareto front using Euclidian distances. The p-center problems, both in the discrete and continuous versions, are proven solvable in polynomial time with a common dynamic programming algorithm. Having $N$ points to partition in $K\geqslant 3$ clusters, the complexity is proven in $O(KN\log N)$ (resp $O(KN\log^2 N)$) time and $O(KN)$ memory space for the continuous (resp discrete) $K$-center problem. $2$-center problems have complexities in $O(N\log N)$. To speed-up the algorithm, parallelization issues are discussed. A posteriori, these results allow an application inside multi-objective heuristics to archive partial Pareto Fronts.
Optimal transport offers an alternative to maximum likelihood for learning generative autoencoding models. We show that minimizing the p-Wasserstein distance between the generator and the true data distribution is equivalent to the unconstrained min-min optimization of the p-Wasserstein distance between the encoder aggregated posterior and the prior in latent space, plus a reconstruction error. We also identify the role of its trade-off hyperparameter as the capacity of the generator: its Lipschitz constant. Moreover, we prove that optimizing the encoder over any class of universal approximators, such as deterministic neural networks, is enough to come arbitrarily close to the optimum. We therefore advertise this framework, which holds for any metric space and prior, as a sweet-spot of current generative autoencoding objectives. We then introduce the Sinkhorn auto-encoder (SAE), which approximates and minimizes the p-Wasserstein distance in latent space via backprogation through the Sinkhorn algorithm. SAE directly works on samples, i.e. it models the aggregated posterior as an implicit distribution, with no need for a reparameterization trick for gradients estimations. SAE is thus able to work with different metric spaces and priors with minimal adaptations. We demonstrate the flexibility of SAE on latent spaces with different geometries and priors and compare with other methods on benchmark data sets.
We report a closed-form expression for the Kullback-Leibler divergence between Cauchy distributions which involves the calculation of a novel definite integral. The formula shows that the Kullback-Leibler divergence between Cauchy densities is always finite and symmetric.
We consider both finite and infinite power chi expansions of $f$-divergences derived from Taylor's expansions of smooth generators, and elaborate on cases where these expansions yield closed-form formula, bounded approximations, or analytic divergence series expressions of $f$-divergences.
The traditional Minkowski distances are induced by the corresponding Minkowski norms in real-valued vector spaces. In this work, we propose novel statistical symmetric distances based on the Minkowski's inequality for probability densities belonging to Lebesgue spaces. These statistical Minkowski distances admit closed-form formula for Gaussian mixture models when parameterized by integer exponents. This result extends to arbitrary mixtures of exponential families with natural parameter spaces being cones: This includes the binomial, the multinomial, the zero-centered Laplacian, the Gaussian and the Wishart mixtures, among others. We also derive a Minkowski's diversity index of a normalized weighted set of probability distributions from Minkowski's inequality.
Separable Bregman divergences induce Riemannian metric spaces that are isometric to the Euclidean space after monotone embeddings. We investigate fixed rate quantization and its codebook Voronoi diagrams, and report on experimental performances of partition-based, hierarchical, and soft clustering algorithms with respect to these Riemann-Bregman distances.
Distances are fundamental primitives whose choice significantly impacts the performances of algorithms in machine learning and signal processing. However selecting the most appropriate distance for a given task is an endeavor. Instead of testing one by one the entries of an ever-expanding dictionary of {\em ad hoc} distances, one rather prefers to consider parametric classes of distances that are exhaustively characterized by axioms derived from first principles. Bregman divergences are such a class. However fine-tuning a Bregman divergence is delicate since it requires to smoothly adjust a functional generator. In this work, we propose an extension of Bregman divergences called the Bregman chord divergences. This new class of distances does not require gradient calculations, uses two scalar parameters that can be easily tailored in applications, and generalizes asymptotically Bregman divergences.
The total variation distance is a core statistical distance between probability measures that satisfies the metric axioms, with value always falling in $[0,1]$. This distance plays a fundamental role in machine learning and signal processing: It is a member of the broader class of $f$-divergences, and it is related to the probability of error in Bayesian hypothesis testing. Since the total variation distance does not admit closed-form expressions for statistical mixtures (like Gaussian mixture models), one often has to rely in practice on costly numerical integrations or on fast Monte Carlo approximations that however do not guarantee deterministic lower and upper bounds. In this work, we consider two methods for bounding the total variation of univariate mixture models: The first method is based on the information monotonicity property of the total variation to design guaranteed nested deterministic lower bounds. The second method relies on computing the geometric lower and upper envelopes of weighted mixture components to derive deterministic bounds based on density ratio. We demonstrate the tightness of our bounds in a series of experiments on Gaussian, Gamma and Rayleigh mixture models.
We propose a new generic type of stochastic neurons, called $q$-neurons, that considers activation functions based on Jackson's $q$-derivatives with stochastic parameters $q$. Our generalization of neural network architectures with $q$-neurons is shown to be both scalable and very easy to implement. We demonstrate experimentally consistently improved performances over state-of-the-art standard activation functions, both on training and testing loss functions.
Exponential families and mixture families are parametric probability models that can be geometrically studied as smooth statistical manifolds with respect to any statistical divergence like the Kullback-Leibler (KL) divergence or the Hellinger divergence. When equipping a statistical manifold with the KL divergence, the induced manifold structure is dually flat, and the KL divergence between distributions amounts to an equivalent Bregman divergence on their corresponding parameters. In practice, the corresponding Bregman generators of mixture/exponential families require to perform definite integral calculus that can either be too time-consuming (for exponentially large discrete support case) or even do not admit closed-form formula (for continuous support case). In these cases, the dually flat construction remains theoretical and cannot be used by information-geometric algorithms. To bypass this problem, we consider performing stochastic Monte Carlo (MC) estimation of those integral-based mixture/exponential family Bregman generators. We show that, under natural assumptions, these MC generators are almost surely Bregman generators. We define a series of dually flat information geometries, termed Monte Carlo Information Geometries, that increasingly-finely approximate the untractable geometry. The advantage of this MCIG is that it allows a practical use of the Bregman algorithmic toolbox on a wide range of probability distribution families. We demonstrate our approach with a clustering task on a mixture family manifold.
In Valiant's model of evolution, a class of representations is evolvable iff a polynomial-time process of random mutations guided by selection converges with high probability to a representation as $ε$-close as desired from the optimal one, for any required $ε>0$. Several previous positive results exist that can be related to evolving a vector space, but each former result imposes disproportionate representations or restrictions on (re)initialisations, distributions, performance functions and/or the mutator. In this paper, we show that all it takes to evolve a normed vector space is merely a set that generates the space. Furthermore, it takes only $\tilde{O}(1/ε^2)$ steps and it is essentially stable, agnostic and handles target drifts that rival some proven in fairly restricted settings. Our algorithm can be viewed as a close relative to a popular fifty-years old gradient-free optimization method for which little is still known from the convergence standpoint: Nelder-Mead simplex method.
We introduce a novel family of distances, called the chord gap divergences, that generalizes the Jensen divergences (also called the Burbea-Rao distances), and study its properties. It follows a generalization of the celebrated statistical Bhattacharyya distance that is frequently met in applications. We report an iterative concave-convex procedure for computing centroids, and analyze the performance of the $k$-means++ clustering with respect to that new dissimilarity measure by introducing the Taylor-Lagrange remainder form of the skew Jensen divergences.
Recurrent Neural Networks (RNNS) are now widely used on sequence generation tasks due to their ability to learn long-range dependencies and to generate sequences of arbitrary length. However, their left-to-right generation procedure only allows a limited control from a potential user which makes them unsuitable for interactive and creative usages such as interactive music generation. This paper introduces a novel architecture called Anticipation-RNN which possesses the assets of the RNN-based generative models while allowing to enforce user-defined positional constraints. We demonstrate its efficiency on the task of generating melodies satisfying positional constraints in the style of the soprano parts of the J.S. Bach chorale harmonizations. Sampling using the Anticipation-RNN is of the same order of complexity than sampling from the traditional RNN model. This fast and interactive generation of musical sequences opens ways to devise real-time systems that could be used for creative purposes.
Distances on symbolic musical sequences are needed for a variety of applications, from music retrieval to automatic music generation. These musical sequences belong to a given corpus (or style) and it is obvious that a good distance on musical sequences should take this information into account; being able to define a distance ex nihilo which could be applicable to all music styles seems implausible. A distance could also be invariant under some transformations, such as transpositions, so that it can be used as a distance between musical motives rather than musical sequences. However, to our knowledge, none of the approaches to devise musical distances seem to address these issues. This paper introduces a method to build transposition-invariant distances on symbolic musical sequences which are learned from data. It is a hybrid distance which combines learned feature representations of musical sequences with a handcrafted rank distance. This distance depends less on the musical encoding of the data than previous methods and gives perceptually good results. We demonstrate its efficiency on the dataset of chorale melodies by J.S. Bach.
VAEs (Variational AutoEncoders) have proved to be powerful in the context of density modeling and have been used in a variety of contexts for creative purposes. In many settings, the data we model possesses continuous attributes that we would like to take into account at generation time. We propose in this paper GLSR-VAE, a Geodesic Latent Space Regularization for the Variational AutoEncoder architecture and its generalizations which allows a fine control on the embedding of the data into the latent space. When augmenting the VAE loss with this regularization, changes in the learned latent space reflects changes of the attributes of the data. This deeper understanding of the VAE latent space structure offers the possibility to modulate the attributes of the generated data in a continuous way. We demonstrate its efficiency on a monophonic music generation task where we manage to generate variations of discrete sequences in an intended and playful way.