Polynomial Neural Networks (PNNs) possess a rich algebraic and geometric structure. However, their identifiability-a key property for ensuring interpretability-remains poorly understood. In this work, we present a comprehensive analysis of the identifiability of deep PNNs, including architectures with and without bias terms. Our results reveal an intricate interplay between activation degrees and layer widths in achieving identifiability. As special cases, we show that architectures with non-increasing layer widths are generically identifiable under mild conditions, while encoder-decoder networks are identifiable when the decoder widths do not grow too rapidly compared to the activation degrees. Our proofs are constructive and center on a connection between deep PNNs and low-rank tensor decompositions, and Kruskal-type uniqueness theorems. We also settle an open conjecture on the dimension of PNN's neurovarieties, and provide new bounds on the activation degrees required for it to reach the expected dimension.

Cluster DAGs (C-DAGs) provide an abstraction of causal graphs in which nodes represent clusters of variables, and edges encode both cluster-level causal relationships and dependencies arisen from unobserved confounding. C-DAGs define an equivalence class of acyclic causal graphs that agree on cluster-level relationships, enabling causal reasoning at a higher level of abstraction. However, when the chosen clustering induces cycles in the resulting C-DAG, the partition is deemed inadmissible under conventional C-DAG semantics. In this work, we extend the C-DAG framework to support arbitrary variable clusterings by relaxing the partition admissibility constraint, thereby allowing cyclic C-DAG representations. We extend the notions of d-separation and causal calculus to this setting, significantly broadening the scope of causal reasoning across clusters and enabling the application of C-DAGs in previously intractable scenarios. Our calculus is both sound and atomically complete with respect to the do-calculus: all valid interventional queries at the cluster level can be derived using our rules, each corresponding to a primitive do-calculus step.

In this article, we present an innovative clustering framework designed for large datasets and real-time data streams which uses a sliding window and histogram model to address the challenge of memory congestion while reducing computational complexity and improving cluster quality for both static and dynamic clustering. The framework provides a simple way to characterize the probability distribution of cluster distributions through histogram models, regardless of their distribution type. This advantage allows for efficient use with various conventional clustering algorithms . To facilitate effective clustering across windows, we use a statistical measure that allows the comparison and merging of different clusters based on the calculation of the Wasserstein distance between histograms.

Abstract. This work introduces polarimetric Fourier phase retrieval (PPR), a physically inspired model to leverage polarization of light information in Fourier phase retrieval problems. We provide a complete characterization of its uniqueness properties by unraveling equivalencies with two related problems, namely, bivariate phase retrieval and a polynomial autocorrelation factorization problem. In particular, we show that the problem admits a unique solution, which can be formulated as a greatest common divisor (GCD) of measurement polynomials. As a result, we propose algebraic solutions for PPR based on approximate GCD computations using the null-space properties of Sylvester matrices. Alternatively, existing iterative algorithms for phase retrieval, semidefinite positive relaxation and Wirtinger flow, are carefully adapted to solve the PPR problem. Finally, a set of numerical experiments permits a detailed assessment of the numerical behavior and relative performances of each proposed reconstruction strategy. They further demonstrate the fruitful combination of algebraic and iterative approaches toward a scalable, computationally efficient, and robust to noise reconstruction strategy for PPR.

This paper discusses the problem of autonomously constructing a qualitative map of an unknown 3D environment using a 3D-Lidar. In this case, how can we effectively integrate the quality of the 3D-reconstruction into the selection of the Next-Best-View? Here, we address the challenge of estimating the quality of the currently reconstructed map in order to guide the exploration policy, in the absence of ground truth, which is typically the case in exploration scenarios. Our key contribution is a method to build a prior on the quality of the reconstruction from the data itself. Indeed, we not only prove that this quality depends on statistics from the observation viewpoints, but we also demonstrate that we can enhance the quality of the reconstruction by leveraging these statistics during the exploration. To do so, we propose to integrate them into Next-Best-View selection policies, in which the information gain is directly computed based on these statistics. Finally, we demonstrate the robustness of our approach, even in challenging environments, with noise in the robot localization, and we further validate it through a real-world experiment.

To address functional-output regression, we introduce projection learning (PL), a novel dictionary-based approach that learns to predict a function that is expanded on a dictionary while minimizing an empirical risk based on a functional loss. PL makes it possible to use non orthogonal dictionaries and can then be combined with dictionary learning; it is thus much more flexible than expansion-based approaches relying on vectorial losses. This general method is instantiated with reproducing kernel Hilbert spaces of vector-valued functions as kernel-based projection learning (KPL). For the functional square loss, two closed-form estimators are proposed, one for fully observed output functions and the other for partially observed ones. Both are backed theoretically by an excess risk analysis. Then, in the more general setting of integral losses based on differentiable ground losses, KPL is implemented using first-order optimization for both fully and partially observed output functions. Eventually, several robustness aspects of the proposed algorithms are highlighted on a toy dataset; and a study on two real datasets shows that they are competitive compared to other nonlinear approaches. Notably, using the square loss and a learnt dictionary, KPL enjoys a particularily attractive trade-off between computational cost and performances.

In this paper, we propose a theoretically supported sequential strategy for training a large-scale Recommender System (RS) over implicit feedback, mainly in the form of clicks. The proposed approach consists in minimizing pairwise ranking loss over blocks of consecutive items constituted by a sequence of non-clicked items followed by a clicked one for each user. We present two variants of this strategy where model parameters are updated using either the momentum method or a gradient-based approach. To prevent updating the parameters for an abnormally high number of clicks over some targeted items (mainly due to bots), we introduce an upper and a lower threshold on the number of updates for each user. These thresholds are estimated over the distribution of the number of blocks in the training set. They affect the decision of RS by shifting the distribution of items that are shown to the users. Furthermore, we provide a convergence analysis of both algorithms and demonstrate their practical efficiency over six large-scale collections with respect to various ranking measures and computational time.

We propose here a generalization of regression trees, referred to as Probabilistic Regression (PR) trees, that adapt to the smoothness of the prediction function relating input and output variables while preserving the interpretability of the prediction and being robust to noise. In PR trees, an observation is associated to all regions of a tree through a probability distribution that reflects how far the observation is to a region. We show that such trees are consistent, meaning that their error tends to 0 when the sample size tends to infinity, a property that has not been established for similar, previous proposals as Soft trees and Smooth Transition Regression trees. We further explain how PR trees can be used in different ensemble methods, namely Random Forests and Gradient Boosted Trees. Lastly, we assess their performance through extensive experiments that illustrate their benefits in terms of performance, interpretability and robustness to noise.

We present a new convex formulation for the problem of recovering lines in degraded images. Following the recent paradigm of superresolution, we formulate a dedicated atomic norm penalty and we solve this optimization problem by means of a primal-dual algorithm. This parsimonious model enables the reconstruction of lines from lowpass measurements, even in presence of a large amount of noise or blur. Furthermore, a Prony method performed on rows and columns of the restored image, provides a spectral estimation of the line parameters, with subpixel accuracy.

Social media has become a major source for analyzing all aspects of daily life. Thanks to dedicated latent topic analysis methods such as the Ailment Topic Aspect Model (ATAM), public health can now be observed on Twitter. In this work, we are interested in using social media to monitor people's health overtime. The use of tweets has several benefits including instantaneous data availability at virtually no cost. Early monitoring of health data is complementary to post-factum studies and enables a range of applications such as measuring behavioral risk factors and triggering health campaigns. We formulate two problems: health transition detection and health transition prediction. We first propose the Temporal Ailment Topic Aspect Model (TM-ATAM), a new latent model dedicated to solving the first problem by capturing transitions that involve health-related topics. TM-ATAM is a non-obvious extension to ATAM that was designed to extract health-related topics. It learns health-related topic transitions by minimizing the prediction error on topic distributions between consecutive posts at different time and geographic granularities. To solve the second problem, we develop T-ATAM, a Temporal Ailment Topic Aspect Model where time is treated as a random variable natively inside ATAM. Our experiments on an 8-month corpus of tweets show that TM-ATAM outperforms TM-LDA in estimating health-related transitions from tweets for different geographic populations. We examine the ability of TM-ATAM to detect transitions due to climate conditions in different geographic regions. We then show how T-ATAM can be used to predict the most important transition and additionally compare T-ATAM with CDC (Center for Disease Control) data and Google Flu Trends.

Hesam Amoualian, Wei Lu, Eric Gaussier, Georgios Balikas, Massih R. Amini, Marianne Clausel. Proceedings of the 55th Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers). 2017.

The exchangeability assumption in topic models like Latent Dirichlet Allocation (LDA) often results in inferring inconsistent topics for the words of text spans like noun-phrases, which are usually expected to be topically coherent. We propose copulaLDA, that extends LDA by integrating part of the text structure to the model and relaxes the conditional independence assumption between the word-specific latent topics given the per-document topic distributions. To this end, we assume that the words of text spans like noun-phrases are topically bound and we model this dependence with copulas. We demonstrate empirically the effectiveness of copulaLDA on both intrinsic and extrinsic evaluation tasks on several publicly available corpora.

We propose in this paper two new models for modeling topic and word-topic dependencies between consecutive documents in document streams. The first model is a direct extension of Latent Dirichlet Allocation model (LDA) and makes use of a Dirichlet distribution to balance the influence of the LDA prior parameters wrt to topic and word-topic distribution of the previous document. The second extension makes use of copulas, which constitute a generic tools to model dependencies between random variables. We rely here on Archimedean copulas, and more precisely on Franck copulas, as they are symmetric and associative and are thus appropriate for exchangeable random variables. Our experiments, conducted on three standard collections that have been used in several studies on topic modeling, show that our proposals outperform previous ones (as dynamic topic models and temporal \LDA), both in terms of perplexity and for tracking similar topics in a document stream.

Probabilistic topic models are generative models that describe the content of documents by discovering the latent topics underlying them. However, the structure of the textual input, and for instance the grouping of words in coherent text spans such as sentences, contains much information which is generally lost with these models. In this paper, we propose sentenceLDA, an extension of LDA whose goal is to overcome this limitation by incorporating the structure of the text in the generative and inference processes. We illustrate the advantages of sentenceLDA by comparing it with LDA using both intrinsic (perplexity) and extrinsic (text classification) evaluation tasks on different text collections.

Social media has become a major source for analyzing all aspects of daily life. Thanks to dedicated latent topic analysis methods such as the Ailment Topic Aspect Model (ATAM), public health can now be observed on Twitter. In this work, we are interested in monitoring people's health over time. Recently, Temporal-LDA (TM?LDA) was proposed for efficiently modeling general-purpose topic transitions over time. In this paper, we propose Temporal Ailment Topic Aspect (TM?ATAM), a new latent model dedicated to capturing transitions that involve health-related topics. TM?ATAM learns topic transition parameters by minimizing the prediction error on topic distributions between consecutive posts at different time and geographic granularities. Our experiments on an 8-month corpus of tweets show that it largely outperforms its predecessors.

We present an optical flow technique in a differential projected framework adapted to local myocardial motion estimation from MR Tagged images. The algorithm is based on the Dual Tree design of Hilbert transform pairs of wavelet bases. Such a design allows one to construct several orientation-sensitive wavelet filters for better analysis of the complex motion of the heart. The complex wavelet transform (CWT) integrates both energy and phase information in the wavelets coefficients for an effective motion estimation. The CWT also provides a high frequency analysis and enjoys a multiresolution aspect that allows multiscale flow estimate. Performances of the algorithm are evaluated on synthetic tagged MRI sequences for both displacement and strain estimation.

This paper presents a new framework for oriented texture modeling. We introduce a new class of Gaussian fields, called Locally Anisotropic Fractional Brownian Fields, with prescribed local orientation at any point. These fields are a local version of a specific class of anisotropic self-similar Gaussian fields with stationary increments. The simulation of such textures is obtained using a new algorithm mixing the tangent field formulation and a turning band method, this latter method having proved its efficiency for generating stationary anisotropic textures. Numerical experiments show the ability of the method for synthesis of textures with prescribed local orientation.

Textures in images can often be well modeled using self-similar processes while they may simultaneously display anisotropy. The present contribution thus aims at studying jointly selfsimilarity and anisotropy by focusing on a specific classical class of Gaussian anisotropic selfsimilar processes. It will be first shown that accurate joint estimates of the anisotropy and selfsimilarity parameters are performed by replacing the standard 2D-discrete wavelet transform with the hyperbolic wavelet transform, which permits the use of different dilation factors along the horizontal and vertical axes. Defining anisotropy requires a reference direction that needs not a priori match the horizontal and vertical axes according to which the images are digitized; this discrepancy defines a rotation angle. Second, we show that this rotation angle can be jointly estimated. Third, a nonparametric bootstrap based procedure is described, which provides confidence intervals in addition to the estimates themselves and enables us to construct an isotropy test procedure, which can be applied to a single texture image. Fourth, the robustness and versatility of the proposed analysis are illustrated by being applied to a large variety of different isotropic and anisotropic self-similar fields. As an illustration, we show that a true anisotropy built-in self-similarity can be disentangled from an isotropic self-similarity to which an anisotropic trend has been superimposed.