Rethinking LDA: moment matching for discrete ICA

We consider moment matching techniques for estimation in Latent Dirichlet Allocation (LDA). By drawing explicit links between LDA and discrete versions of independent component analysis (ICA), we first derive a new set of cumulant-based tensors, with an improved sample complexity. Moreover, we reuse standard ICA techniques such as joint diagonalization of tensors to improve over existing methods based on the tensor power method. In an extensive set of experiments on both synthetic and real datasets, we show that our new combination of tensors and orthogonal joint diagonalization techniques outperforms existing moment matching methods.Comment: 30 pages; added plate diagrams and clarifications, changed style, corrected typos, updated figures. in Proceedings of the 29-th Conference on Neural Information Processing Systems (NIPS), 201

Overcomplete Independent Component Analysis via SDP

We present a novel algorithm for overcomplete independent components analysis (ICA), where the number of latent sources k exceeds the dimension p of observed variables. Previous algorithms either suffer from high computational complexity or make strong assumptions about the form of the mixing matrix. Our algorithm does not make any sparsity assumption yet enjoys favorable computational and theoretical properties. Our algorithm consists of two main steps: (a) estimation of the Hessians of the cumulant generating function (as opposed to the fourth and higher order cumulants used by most algorithms) and (b) a novel semi-definite programming (SDP) relaxation for recovering a mixing component. We show that this relaxation can be efficiently solved with a projected accelerated gradient descent method, which makes the whole algorithm computationally practical. Moreover, we conjecture that the proposed program recovers a mixing component at the rate k < p^2/4 and prove that a mixing component can be recovered with high probability when k < (2 - epsilon) p log p when the original components are sampled uniformly at random on the hyper sphere. Experiments are provided on synthetic data and the CIFAR-10 dataset of real images.Comment: Appears in: Proceedings of the 22nd International Conference on Artificial Intelligence and Statistics (AISTATS 2019). 21 page

On the method of moments for estimation in latent linear models

Les modèles linéaires latents sont des modèles statistique puissants pour extraire la structure latente utile à partir de données non structurées par ailleurs. Ces modèles sont utiles dans de nombreuses applications telles que le traitement automatique du langage naturel et la vision artificielle. Pourtant, l'estimation et l'inférence sont souvent impossibles en temps polynomial pour de nombreux modèles linéaires latents et on doit utiliser des méthodes approximatives pour lesquelles il est difficile de récupérer les paramètres. Plusieurs approches, introduites récemment, utilisent la méthode des moments. Elles permettent de retrouver les paramètres dans le cadre idéalisé d'un échantillon de données infini tiré selon certains modèles, mais ils viennent souvent avec des garanties théoriques dans les cas où ce n'est pas exactement satisfait. Dans cette thèse, nous nous concentrons sur les méthodes d'estimation fondées sur l'appariement de moment pour différents modèles linéaires latents. L'utilisation d'un lien étroit avec l'analyse en composantes indépendantes, qui est un outil bien étudié par la communauté du traitement du signal, nous présentons plusieurs modèles semiparamétriques pour la modélisation thématique et dans un contexte multi-vues. Nous présentons des méthodes à base de moment ainsi que des algorithmes pour l'estimation dans ces modèles, et nous prouvons pour ces méthodes des résultats de complexité améliorée par rapport aux méthodes existantes. Nous donnons également des garanties d'identifiabilité, contrairement à d'autres modèles actuels. C'est une propriété importante pour assurer leur interprétabilité.Latent linear models are powerful probabilistic tools for extracting useful latent structure from otherwise unstructured data and have proved useful in numerous applications such as natural language processing and computer vision. However, the estimation and inference are often intractable for many latent linear models and one has to make use of approximate methods often with no recovery guarantees. An alternative approach, which has been popular lately, are methods based on the method of moments. These methods often have guarantees of exact recovery in the idealized setting of an infinite data sample and well specified models, but they also often come with theoretical guarantees in cases where this is not exactly satisfied. In this thesis, we focus on moment matchingbased estimation methods for different latent linear models. Using a close connection with independent component analysis, which is a well studied tool from the signal processing literature, we introduce several semiparametric models in the topic modeling context and for multi-view models and develop moment matching-based methods for the estimation in these models. These methods come with improved sample complexity results compared to the previously proposed methods. The models are supplemented with the identifiability guarantees, which is a necessary property to ensure their interpretability. This is opposed to some other widely used models, which are unidentifiable

Sur la méthode des moments pour l'estimation des modèles à variables latentes

Latent linear models are powerful probabilistic tools for extracting useful latent structure from otherwise unstructured data and have proved useful in numerous applications such as natural language processing and computer vision. However, the estimation and inference are often intractable for many latent linear models and one has to make use of approximate methods often with no recovery guarantees. An alternative approach, which has been popular lately, are methods based on the method of moments. These methods often have guarantees of exact recovery in the idealized setting of an infinite data sample and well specified models, but they also often come with theoretical guarantees in cases where this is not exactly satisfied. In this thesis, we focus on moment matchingbased estimation methods for different latent linear models. Using a close connection with independent component analysis, which is a well studied tool from the signal processing literature, we introduce several semiparametric models in the topic modeling context and for multi-view models and develop moment matching-based methods for the estimation in these models. These methods come with improved sample complexity results compared to the previously proposed methods. The models are supplemented with the identifiability guarantees, which is a necessary property to ensure their interpretability. This is opposed to some other widely used models, which are unidentifiable.Les modèles linéaires latents sont des modèles statistique puissants pour extraire la structure latente utile à partir de données non structurées par ailleurs. Ces modèles sont utiles dans de nombreuses applications telles que le traitement automatique du langage naturel et la vision artificielle. Pourtant, l'estimation et l'inférence sont souvent impossibles en temps polynomial pour de nombreux modèles linéaires latents et on doit utiliser des méthodes approximatives pour lesquelles il est difficile de récupérer les paramètres. Plusieurs approches, introduites récemment, utilisent la méthode des moments. Elles permettent de retrouver les paramètres dans le cadre idéalisé d'un échantillon de données infini tiré selon certains modèles, mais ils viennent souvent avec des garanties théoriques dans les cas où ce n'est pas exactement satisfait. Dans cette thèse, nous nous concentrons sur les méthodes d'estimation fondées sur l'appariement de moment pour différents modèles linéaires latents. L'utilisation d'un lien étroit avec l'analyse en composantes indépendantes, qui est un outil bien étudié par la communauté du traitement du signal, nous présentons plusieurs modèles semiparamétriques pour la modélisation thématique et dans un contexte multi-vues. Nous présentons des méthodes à base de moment ainsi que des algorithmes pour l'estimation dans ces modèles, et nous prouvons pour ces méthodes des résultats de complexité améliorée par rapport aux méthodes existantes. Nous donnons également des garanties d'identifiabilité, contrairement à d'autres modèles actuels. C'est une propriété importante pour assurer leur interprétabilité

Beyond CCA: Moment Matching for Multi-View Models

We introduce three novel semi-parametric extensions of probabilistic canonical correlation analysis with identifiability guarantees. We consider moment matching techniques for estimation in these models. For that, by drawing explicit links between the new models and a discrete version of independent component analysis (DICA), we first extend the DICA cumulant tensors to the new discrete version of CCA. By further using a close connection with independent component analysis, we introduce generalized covariance matrices , which can replace the cumulant tensors in the moment matching framework, and, therefore, improve sample complexity and simplify derivations and algorithms significantly. As the tensor power method or orthogonal joint diagonalization are not applicable in the new setting, we use non-orthogonal joint diago-nalization techniques for matching the cumulants. We demonstrate performance of the proposed models and estimation techniques on experiments with both synthetic and real datasets

Overcomplete Independent Component Analysis via SDP

Appears in: Proceedings of the 22nd International Conference on Artificial Intelligence and Statistics (AISTATS 2019). 21 pagesInternational audienceWe present a novel algorithm for overcomplete independent components analysis (ICA), where the number of latent sources k exceeds the dimension p of observed variables. Previous algorithms either suffer from high computational complexity or make strong assumptions about the form of the mixing matrix. Our algorithm does not make any sparsity assumption yet enjoys favorable computational and theoretical properties. Our algorithm consists of two main steps: (a) estimation of the Hessians of the cumulant generating function (as opposed to the fourth and higher order cumulants used by most algorithms) and (b) a novel semi-definite programming (SDP) relaxation for recovering a mixing component. We show that this relaxation can be efficiently solved with a projected accelerated gradient descent method, which makes the whole algorithm computationally practical. Moreover, we conjecture that the proposed program recovers a mixing component at the rate k < p^2/4 and prove that a mixing component can be recovered with high probability when k < (2 - epsilon) p log p when the original components are sampled uniformly at random on the hyper sphere. Experiments are provided on synthetic data and the CIFAR-10 dataset of real images

Overcomplete Independent Component Analysis via SDP

Appears in: Proceedings of the 22nd International Conference on Artificial Intelligence and Statistics (AISTATS 2019). 21 pagesInternational audienceWe present a novel algorithm for overcomplete independent components analysis (ICA), where the number of latent sources k exceeds the dimension p of observed variables. Previous algorithms either suffer from high computational complexity or make strong assumptions about the form of the mixing matrix. Our algorithm does not make any sparsity assumption yet enjoys favorable computational and theoretical properties. Our algorithm consists of two main steps: (a) estimation of the Hessians of the cumulant generating function (as opposed to the fourth and higher order cumulants used by most algorithms) and (b) a novel semi-definite programming (SDP) relaxation for recovering a mixing component. We show that this relaxation can be efficiently solved with a projected accelerated gradient descent method, which makes the whole algorithm computationally practical. Moreover, we conjecture that the proposed program recovers a mixing component at the rate k < p^2/4 and prove that a mixing component can be recovered with high probability when k < (2 - epsilon) p log p when the original components are sampled uniformly at random on the hyper sphere. Experiments are provided on synthetic data and the CIFAR-10 dataset of real images

Overcomplete independent component analysis via SDP

We present a novel algorithm for overcomplete independent components analysis (ICA), where the number of latent sources k exceeds the dimension p of observed variables. Previous algorithms either suffer from high computational complexity or make strong assumptions about the form of the mixing matrix. Our algorithm does not make any sparsity assumption yet enjoys favorable computational and theoretical properties. Our algorithm consists of two main steps: (a) estimation of the Hessians of the cumulant generating function (as opposed to the fourth and higher order cumulants used by most algorithms) and (b) a novel semi-definite programming (SDP) relaxation for recovering a mixing component. We show that this relaxation can be efficiently solved with a projected accelerated gradient descent method, which makes the whole algorithm computationally practical. Moreover, we conjecture that the proposed program recovers a mixing component at the rate k < p2/4 and prove that a mixing component can be recovered with high probability when k < (2 - ")plog p when the original components are sampled uniformly at random on the hypersphere. Experiments are provided on synthetic data and the CIFAR-10 dataset of real images.United States. Defense Advanced Research Projects Agency (Grant W911NF-16-1-0551)National Science Foundation (U.S.). Career Grant (Awards 1350965, CCF-1453261)National Science Foundation (U.S.) (Grant DMS-1712730