Ronen Talmon

Tal Shnitzer, Ronen Talmon, Jean-Jacques Slotine

Analyzing signals arising from dynamical systems typically requires many modeling assumptions and parameter estimation. In high dimensions, this modeling is particularly difficult due to the "curse of dimensionality". In this paper, we propose a method for building an intrinsic representation of such signals in a purely data-driven manner. First, we apply a manifold learning technique, diffusion maps, to learn the intrinsic model of the latent variables of the dynamical system, solely from the measurements. Second, we use concepts and tools from control theory and build a linear contracting observer to estimate the latent variables in a sequential manner from new incoming measurements. The effectiveness of the presented framework is demonstrated by applying it to a toy problem and to a music analysis application. In these examples we show that our method reveals the intrinsic variables of the analyzed dynamical systems.

David Cohen, Tal Shnitzer, Yuval Kluger et al.

In this paper, we present a new method for few-sample supervised feature selection (FS). Our method first learns the manifold of the feature space of each class using kernels capturing multi-feature associations. Then, based on Riemannian geometry, a composite kernel is computed, extracting the differences between the learned feature associations. Finally, a FS score based on spectral analysis is proposed. Considering multi-feature associations makes our method multivariate by design. This in turn allows for the extraction of the hidden manifold underlying the features and avoids overfitting, facilitating few-sample FS. We showcase the efficacy of our method on illustrative examples and several benchmarks, where our method demonstrates higher accuracy in selecting the informative features compared to competing methods. In addition, we show that our FS leads to improved classification and better generalization when applied to test data.

Tal Shnitzer, Ronen Talmon, Jean-Jacques Slotine

In this paper, we propose a non-parametric method for state estimation of high-dimensional nonlinear stochastic dynamical systems, which evolve according to gradient flows with isotropic diffusion. We combine diffusion maps, a manifold learning technique, with a linear Kalman filter and with concepts from Koopman operator theory. More concretely, using diffusion maps, we construct data-driven virtual state coordinates, which linearize the system model. Based on these coordinates, we devise a data-driven framework for state estimation using the Kalman filter. We demonstrate the strengths of our method with respect to both parametric and non-parametric algorithms in three tracking problems. In particular, applying the approach to actual recordings of hippocampal neural activity in rodents directly yields a representation of the position of the animals. We show that the proposed method outperforms competing non-parametric algorithms in the examined stochastic problem formulations. Additionally, we obtain results comparable to classical parametric algorithms, which, in contrast to our method, are equipped with model knowledge.

Adi Arbel, Stefan Steinerberger, Ronen Talmon

Given two symmetric positive-definite matrices $A, B \in \mathbb{R}^{n \times n}$, we study the spectral properties of the interpolation $A^{1-x} B^x$ for $0 \leq x \leq 1$. The presence of `common structures' in $A$ and $B$, eigenvectors pointing in a similar direction, can be investigated using this interpolation perspective. Generically, exact log-linearity of the operator norm $\|A^{1-x} B^x\|$ is equivalent to the existence of a shared eigenvector in the original matrices; stability bounds show that approximate log-linearity forces principal singular vectors to align with leading eigenvectors of both matrices. These results give rise to and provide theoretical justification for a multi-manifold learning framework that identifies common and distinct latent structures in multiview data.

Ya-Wei Eileen Lin, Ronald R. Coifman, Gal Mishne et al.

Finding meaningful distances between high-dimensional data samples is an important scientific task. To this end, we propose a new tree-Wasserstein distance (TWD) for high-dimensional data with two key aspects. First, our TWD is specifically designed for data with a latent feature hierarchy, i.e., the features lie in a hierarchical space, in contrast to the usual focus on embedding samples in hyperbolic space. Second, while the conventional use of TWD is to speed up the computation of the Wasserstein distance, we use its inherent tree as a means to learn the latent feature hierarchy. The key idea of our method is to embed the features into a multi-scale hyperbolic space using diffusion geometry and then present a new tree decoding method by establishing analogies between the hyperbolic embedding and trees. We show that our TWD computed based on data observations provably recovers the TWD defined with the latent feature hierarchy and that its computation is efficient and scalable. We showcase the usefulness of the proposed TWD in applications to word-document and single-cell RNA-sequencing datasets, demonstrating its advantages over existing TWDs and methods based on pre-trained models.

Thomas Dagès, Simon Weber, Ya-Wei Eileen Lin et al.

Dimensionality reduction is a fundamental task that aims to simplify complex data by reducing its feature dimensionality while preserving essential patterns, with core applications in data analysis and visualisation. To preserve the underlying data structure, multi-dimensional scaling (MDS) methods focus on preserving pairwise dissimilarities, such as distances. They optimise the embedding to have pairwise distances as close as possible to the data dissimilarities. However, the current standard is limited to embedding data in Riemannian manifolds. Motivated by the lack of asymmetry in the Riemannian metric of the embedding space, this paper extends the MDS problem to a natural asymmetric generalisation of Riemannian manifolds called Finsler manifolds. Inspired by Euclidean space, we define a canonical Finsler space for embedding asymmetric data. Due to its simplicity with respect to geodesics, data representation in this space is both intuitive and simple to analyse. We demonstrate that our generalisation benefits from the same theoretical convergence guarantees. We reveal the effectiveness of our Finsler embedding across various types of non-symmetric data, highlighting its value in applications such as data visualisation, dimensionality reduction, directed graph embedding, and link prediction.

Ya-Wei Eileen Lin, Ronald R. Coifman, Gal Mishne et al.

High-dimensional data often exhibit hierarchical structures in both modes: samples and features. Yet, most existing approaches for hierarchical representation learning consider only one mode at a time. In this work, we propose an unsupervised method for jointly learning hierarchical representations of samples and features via Tree-Wasserstein Distance (TWD). Our method alternates between the two data modes. It first constructs a tree for one mode, then computes a TWD for the other mode based on that tree, and finally uses the resulting TWD to build the second mode's tree. By repeatedly alternating through these steps, the method gradually refines both trees and the corresponding TWDs, capturing meaningful hierarchical representations of the data. We provide a theoretical analysis showing that our method converges. We show that our method can be integrated into hyperbolic graph convolutional networks as a pre-processing technique, improving performance in link prediction and node classification tasks. In addition, our method outperforms baselines in sparse approximation and unsupervised Wasserstein distance learning tasks on word-document and single-cell RNA-sequencing datasets.

Sing-Yuan Yeh, Hau-Tieng Wu, Ronen Talmon et al.

Alternating Diffusion (AD) is a commonly applied diffusion-based sensor fusion algorithm. While it has been successfully applied to various problems, its computational burden remains a limitation. Inspired by the landmark diffusion idea considered in the Robust and Scalable Embedding via Landmark Diffusion (ROSELAND), we propose a variation of AD, called Landmark AD (LAD), which captures the essence of AD while offering superior computational efficiency. We provide a series of theoretical analyses of LAD under the manifold setup and apply it to the automatic sleep stage annotation problem with two electroencephalogram channels to demonstrate its application.

Idan Simai, Ronen Talmon, Uri Shaham

In this paper, we argue that viewing VICReg-a popular self-supervised learning (SSL) method--through the lens of spectral embedding reveals a potential source of sub-optimality: it may struggle to generalize robustly to unseen data due to overreliance on the training data. This observation invites a closer look at how well this method achieves its goal of producing meaningful representations of images outside of the training set as well. Here, we investigate this issue and introduce SAG-VICReg (Stable and Generalizable VICReg), a method that builds on VICReg by incorporating new training techniques. These enhancements improve the model's ability to capture global semantics within the data and strengthen the generalization capabilities. Experiments demonstrate that SAG-VICReg effectively addresses the generalization challenge while matching or surpassing diverse state-of-the-art SSL baselines. Notably, our method exhibits superior performance on metrics designed to evaluate global semantic understanding, while simultaneously maintaining competitive results on local evaluation metrics. Furthermore, we propose a new standalone evaluation metric for embeddings that complements the standard evaluation methods and accounts for the global data structure without requiring labels--a key issue when tagged data is scarce or not available.

Amitai Yacobi, Nir Ben-Ari, Ronen Talmon et al.

Learning shared representations is a primary area of multimodal representation learning. The current approaches to achieve a shared embedding space rely heavily on paired samples from each modality, which are significantly harder to obtain than unpaired ones. In this work, we demonstrate that shared representations can be learned almost exclusively from unpaired data. Our arguments are grounded in the spectral embeddings of the random walk matrices constructed independently from each unimodal representation. Empirical results in computer vision and natural language processing domains support its potential, revealing the effectiveness of unpaired data in capturing meaningful cross-modal relations, demonstrating high capabilities in retrieval tasks, generation, arithmetics, zero-shot, and cross-domain classification. This work, to the best of our knowledge, is the first to demonstrate these capabilities almost exclusively from unpaired samples, giving rise to a cross-modal embedding that could be viewed as universal, i.e., independent of the specific modalities of the data. Our project page: https://shaham-lab.github.io/SUE_page.

Harel Mendelman, Haggai Maron, Ronen Talmon

Graph Neural Networks (GNNs) excel at analyzing graph-structured data but struggle on heterophilic graphs, where connected nodes often belong to different classes. While this challenge is commonly addressed with specialized GNN architectures, graph rewiring remains an underexplored strategy in this context. We provide theoretical foundations linking edge homophily, GNN embedding smoothness, and node classification performance, motivating the need to enhance homophily. Building on this insight, we introduce a rewiring framework that increases graph homophily using a reference graph, with theoretical guarantees on the homophily of the rewired graph. To broaden applicability, we propose a label-driven diffusion approach for constructing a homophilic reference graph from node features and training labels. Through extensive simulations, we analyze how the homophily of both the original and reference graphs influences the rewired graph homophily and downstream GNN performance. We evaluate our method on 11 real-world heterophilic datasets and show that it outperforms existing rewiring techniques and specialized GNNs for heterophilic graphs, achieving improved node classification accuracy while remaining efficient and scalable to large graphs.

David W. Sroczynski, Felix Dietrich, Eleni D. Koronaki et al.

Before we attempt to learn a function between two (sets of) observables of a physical process, we must first decide what the inputs and what the outputs of the desired function are going to be. Here we demonstrate two distinct, data-driven ways of initially deciding ``the right quantities'' to relate through such a function, and then proceed to learn it. This is accomplished by processing multiple simultaneous heterogeneous data streams (ensembles of time series) from observations of a physical system: multiple observation processes of the system. We thus determine (a) what subsets of observables are common between the observation processes (and therefore observable from each other, relatable through a function); and (b) what information is unrelated to these common observables, and therefore particular to each observation process, and not contributing to the desired function. Any data-driven function approximation technique can subsequently be used to learn the input-output relation, from k-nearest neighbors and Geometric Harmonics to Gaussian Processes and Neural Networks. Two particular ``twists'' of the approach are discussed. The first has to do with the identifiability of particular quantities of interest from the measurements. We now construct mappings from a single set of observations of one process to entire level sets of measurements of the process, consistent with this single set. The second attempts to relate our framework to a form of causality: if one of the observation processes measures ``now'', while the second observation process measures ``in the future'', the function to be learned among what is common across observation processes constitutes a dynamical model for the system evolution.

Ya-Wei Eileen Lin, Ronen Talmon, Ron Levie

Equivariant machine learning is an approach for designing deep learning models that respect the symmetries of the problem, with the aim of reducing model complexity and improving generalization. In this paper, we focus on an extension of shift equivariance, which is the basis of convolution networks on images, to general graphs. Unlike images, graphs do not have a natural notion of domain translation. Therefore, we consider the graph functional shifts as the symmetry group: the unitary operators that commute with the graph shift operator. Notably, such symmetries operate in the signal space rather than directly in the spatial space. We remark that each linear filter layer of a standard spectral graph neural network (GNN) commutes with graph functional shifts, but the activation function breaks this symmetry. Instead, we propose nonlinear spectral filters (NLSFs) that are fully equivariant to graph functional shifts and show that they have universal approximation properties. The proposed NLSFs are based on a new form of spectral domain that is transferable between graphs. We demonstrate the superior performance of NLSFs over existing spectral GNNs in node and graph classification benchmarks.

Amitay Bar, Rotem Mulayoff, Tomer Michaeli et al.

Langevin dynamics (LD) is widely used for sampling from distributions and for optimization. In this work, we derive a closed-form expression for the expected loss of preconditioned LD near stationary points of the objective function. We use the fact that at the vicinity of such points, LD reduces to an Ornstein-Uhlenbeck process, which is amenable to convenient mathematical treatment. Our analysis reveals that when the preconditioning matrix satisfies a particular relation with respect to the noise covariance, LD's expected loss becomes proportional to the rank of the objective's Hessian. We illustrate the applicability of this result in the context of neural networks, where the Hessian rank has been shown to capture the complexity of the predictor function but is usually computationally hard to probe. Finally, we use our analysis to compare SGD-like and Adam-like preconditioners and identify the regimes under which each of them leads to a lower expected loss.

Ya-Wei Eileen Lin, Ronald R. Coifman, Gal Mishne et al.

Finding meaningful representations and distances of hierarchical data is important in many fields. This paper presents a new method for hierarchical data embedding and distance. Our method relies on combining diffusion geometry, a central approach to manifold learning, and hyperbolic geometry. Specifically, using diffusion geometry, we build multi-scale densities on the data, aimed to reveal their hierarchical structure, and then embed them into a product of hyperbolic spaces. We show theoretically that our embedding and distance recover the underlying hierarchical structure. In addition, we demonstrate the efficacy of the proposed method and its advantages compared to existing methods on graph embedding benchmarks and hierarchical datasets.

Tal Shnitzer, Hau-Tieng Wu, Ronen Talmon

Multivariate time-series have become abundant in recent years, as many data-acquisition systems record information through multiple sensors simultaneously. In this paper, we assume the variables pertain to some geometry and present an operator-based approach for spatiotemporal analysis. Our approach combines three components that are often considered separately: (i) manifold learning for building operators representing the geometry of the variables, (ii) Riemannian geometry of symmetric positive-definite matrices for multiscale composition of operators corresponding to different time samples, and (iii) spectral analysis of the composite operators for extracting different dynamic modes. We propose a method that is analogous to the classical wavelet analysis, which we term Riemannian multi-resolution analysis (RMRA). We provide some theoretical results on the spectral analysis of the composite operators, and we demonstrate the proposed method on simulations and on real data.

Uri Shaham, Jonathan Svirsky, Ori Katz et al.

Latent variable discovery is a central problem in data analysis with a broad range of applications in applied science. In this work, we consider data given as an invertible mixture of two statistically independent components and assume that one of the components is observed while the other is hidden. Our goal is to recover the hidden component. For this purpose, we propose an autoencoder equipped with a discriminator. Unlike the standard nonlinear ICA problem, which was shown to be non-identifiable, in the special case of ICA we consider here, we show that our approach can recover the component of interest up to entropy-preserving transformation. We demonstrate the performance of the proposed approach in several tasks, including image synthesis, voice cloning, and fetal ECG extraction.

Panagiotis Papaioannou, Ronen Talmon, Ioannis Kevrekidis et al.

We address a three-tier numerical framework based on manifold learning for the forecasting of high-dimensional time series. At the first step, we embed the time series into a reduced low-dimensional space using a nonlinear manifold learning algorithm such as Locally Linear Embedding and Diffusion Maps. At the second step, we construct reduced-order regression models on the manifold, in particular Multivariate Autoregressive (MVAR) and Gaussian Process Regression (GPR) models, to forecast the embedded dynamics. At the final step, we lift the embedded time series back to the original high-dimensional space using Radial Basis Functions interpolation and Geometric Harmonics. For our illustrations, we test the forecasting performance of the proposed numerical scheme with four sets of time series: three synthetic stochastic ones resembling EEG signals produced from linear and nonlinear stochastic models with different model orders, and one real-world data set containing daily time series of 10 key foreign exchange rates (FOREX) spanning the time period 03/09/2001-29/10/2020. The forecasting performance of the proposed numerical scheme is assessed using the combinations of manifold learning, modelling and lifting approaches. We also provide a comparison with the Principal Component Analysis algorithm as well as with the naive random walk model and the MVAR and GPR models trained and implemented directly in the high-dimensional space.

Lior Aloni, Omer Bobrowski, Ronen Talmon

In a world abundant with diverse data arising from complex acquisition techniques, there is a growing need for new data analysis methods. In this paper we focus on high-dimensional data that are organized into several hierarchical datasets. We assume that each dataset consists of complex samples, and every sample has a distinct irregular structure modeled by a graph. The main novelty in this work lies in the combination of two complementing powerful data-analytic approaches: topological data analysis (TDA) and geometric manifold learning. Geometry primarily contains local information, while topology inherently provides global descriptors. Based on this combination, we present a method for building an informative representation of hierarchical datasets. At the finer (sample) level, we devise a new metric between samples based on manifold learning that facilitates quantitative structural analysis. At the coarser (dataset) level, we employ TDA to extract qualitative structural information from the datasets. We showcase the applicability and advantages of our method on simulated data and on a corpus of hyper-spectral images. We show that an ensemble of hyper-spectral images exhibits a hierarchical structure that fits well the considered setting. In addition, we show that our new method gives rise to superior classification results compared to state-of-the-art methods.

Ori Katz, Roy R. Lederman, Ronen Talmon

In this paper, we consider data acquired by multimodal sensors capturing complementary aspects and features of a measured phenomenon. We focus on a scenario in which the measurements share mutual sources of variability but might also be contaminated by other measurement-specific sources such as interferences or noise. Our approach combines manifold learning, which is a class of nonlinear data-driven dimension reduction methods, with the well-known Riemannian geometry of symmetric and positive-definite (SPD) matrices. Manifold learning typically includes the spectral analysis of a kernel built from the measurements. Here, we take a different approach, utilizing the Riemannian geometry of the kernels. In particular, we study the way the spectrum of the kernels changes along geodesic paths on the manifold of SPD matrices. We show that this change enables us, in a purely unsupervised manner, to derive a compact, yet informative, description of the relations between the measurements, in terms of their underlying components. Based on this result, we present new algorithms for extracting the common latent components and for identifying common and measurement-specific components.

Ofir Lindenbaum, Amir Sagiv, Gal Mishne et al.

A low-dimensional dynamical system is observed in an experiment as a high-dimensional signal; for example, a video of a chaotic pendulums system. Assuming that we know the dynamical model up to some unknown parameters, can we estimate the underlying system's parameters by measuring its time-evolution only once? The key information for performing this estimation lies in the temporal inter-dependencies between the signal and the model. We propose a kernel-based score to compare these dependencies. Our score generalizes a maximum likelihood estimator for a linear model to a general nonlinear setting in an unknown feature space. We estimate the system's underlying parameters by maximizing the proposed score. We demonstrate the accuracy and efficiency of the method using two chaotic dynamical systems - the double pendulum and the Lorenz '63 model.

Or Yair, Almog Lahav, Ronen Talmon

In this paper, we present new results on the Riemannian geometry of symmetric positive semi-definite (SPSD) matrices. First, based on an existing approximation of the geodesic path, we introduce approximations of the logarithmic and exponential maps. Second, we present a closed-form expression for Parallel Transport (PT). Third, we derive a canonical representation for a set of SPSD matrices. Based on these results, we propose an algorithm for Domain Adaptation (DA) and demonstrate its performance in two applications: fusion of hyper-spectral images and motion identification.

Felix Dietrich, Or Yair, Rotem Mulayoff et al.

In this paper, we propose a spectral method for deriving functions that are jointly smooth on multiple observed manifolds. This allows us to register measurements of the same phenomenon by heterogeneous sensors, and to reject sensor-specific noise. Our method is unsupervised and primarily consists of two steps. First, using kernels, we obtain a subspace spanning smooth functions on each separate manifold. Then, we apply a spectral method to the obtained subspaces and discover functions that are jointly smooth on all manifolds. We show analytically that our method is guaranteed to provide a set of orthogonal functions that are as jointly smooth as possible, ordered by increasing Dirichlet energy from the smoothest to the least smooth. In addition, we show that the extracted functions can be efficiently extended to unseen data using the Nyström method. We demonstrate the proposed method on both simulated and real measured data and compare the results to nonlinear variants of the seminal Canonical Correlation Analysis (CCA). Particularly, we show superior results for sleep stage identification. In addition, we show how the proposed method can be leveraged for finding minimal realizations of parameter spaces of nonlinear dynamical systems.

Amitay Bar, Ronen Talmon, Ron Meir

Options have been shown to be an effective tool in reinforcement learning, facilitating improved exploration and learning. In this paper, we present an approach based on spectral graph theory and derive an algorithm that systematically discovers options without access to a specific reward or task assignment. As opposed to the common practice used in previous methods, our algorithm makes full use of the spectrum of the graph Laplacian. Incorporating modes associated with higher graph frequencies unravels domain subtleties, which are shown to be useful for option discovery. Using geometric and manifold-based analysis, we present a theoretical justification for the algorithm. In addition, we showcase its performance in several domains, demonstrating clear improvements compared to competing methods.

Or Yair, Felix Dietrich, Ronen Talmon et al.

In this paper, we address the problem of Domain Adaptation (DA) using Optimal Transport (OT) on Riemannian manifolds. We model the difference between two domains by a diffeomorphism and use the polar factorization theorem to claim that OT is indeed optimal for DA in a well-defined sense, up to a volume preserving map. We then focus on the manifold of Symmetric and Positive-Definite (SPD) matrices, whose structure provided a useful context in recent applications. We demonstrate the polar factorization theorem on this manifold. Due to the uniqueness of the weighted Riemannian mean, and by exploiting existing regularized OT algorithms, we formulate a simple algorithm that maps the source domain to the target domain. We test our algorithm on two Brain-Computer Interface (BCI) data sets and observe state of the art performance.

Ariel Schwartz, Ronen Talmon

Data living on manifolds commonly appear in many applications. Often this results from an inherently latent low-dimensional system being observed through higher dimensional measurements. We show that under certain conditions, it is possible to construct an intrinsic and isometric data representation, which respects an underlying latent intrinsic geometry. Namely, we view the observed data only as a proxy and learn the structure of a latent unobserved intrinsic manifold, whereas common practice is to learn the manifold of the observed data. For this purpose, we build a new metric and propose a method for its robust estimation by assuming mild statistical priors and by using artificial neural networks as a mechanism for metric regularization and parametrization. We show successful application to unsupervised indoor localization in ad-hoc sensor networks. Specifically, we show that our proposed method facilitates accurate localization of a moving agent from imaging data it collects. Importantly, our method is applied in the same way to two different imaging modalities, thereby demonstrating its intrinsic and modality-invariant capabilities.

Bracha Laufer-Goldshtein, Ronen Talmon, Sharon Gannot

Blind source separation (BSS) is addressed, using a novel data-driven approach, based on a well-established probabilistic model. The proposed method is specifically designed for separation of multichannel audio mixtures. The algorithm relies on spectral decomposition of the correlation matrix between different time frames. The probabilistic model implies that the column space of the correlation matrix is spanned by the probabilities of the various speakers across time. The number of speakers is recovered by the eigenvalue decay, and the eigenvectors form a simplex of the speakers' probabilities. Time frames dominated by each of the speakers are identified exploiting convex geometry tools on the recovered simplex. The mixing acoustic channels are estimated utilizing the identified sets of frames, and a linear umixing is performed to extract the individual speakers. The derived simplexes are visually demonstrated for mixtures of 2, 3 and 4 speakers. We also conduct a comprehensive experimental study, showing high separation capabilities in various reverberation conditions.

Gautam Pai, Ronen Talmon, Alex Bronstein et al.

This paper explores a fully unsupervised deep learning approach for computing distance-preserving maps that generate low-dimensional embeddings for a certain class of manifolds. We use the Siamese configuration to train a neural network to solve the problem of least squares multidimensional scaling for generating maps that approximately preserve geodesic distances. By training with only a few landmarks, we show a significantly improved local and nonlocal generalization of the isometric mapping as compared to analogous non-parametric counterparts. Importantly, the combination of a deep-learning framework with a multidimensional scaling objective enables a numerical analysis of network architectures to aid in understanding their representation power. This provides a geometric perspective to the generalizability of deep learning.

Gal Mishne, Ronen Talmon, Israel Cohen et al.

We consider the analysis of high dimensional data given in the form of a matrix with columns consisting of observations and rows consisting of features. Often the data is such that the observations do not reside on a regular grid, and the given order of the features is arbitrary and does not convey a notion of locality. Therefore, traditional transforms and metrics cannot be used for data organization and analysis. In this paper, our goal is to organize the data by defining an appropriate representation and metric such that they respect the smoothness and structure underlying the data. We also aim to generalize the joint clustering of observations and features in the case the data does not fall into clear disjoint groups. For this purpose, we propose multiscale data-driven transforms and metrics based on trees. Their construction is implemented in an iterative refinement procedure that exploits the co-dependencies between features and observations. Beyond the organization of a single dataset, our approach enables us to transfer the organization learned from one dataset to another and to integrate several datasets together. We present an application to breast cancer gene expression analysis: learning metrics on the genes to cluster the tumor samples into cancer sub-types and validating the joint organization of both the genes and the samples. We demonstrate that using our approach to combine information from multiple gene expression cohorts, acquired by different profiling technologies, improves the clustering of tumor samples.

Almog Lahav, Ronen Talmon, Yuval Kluger

A fundamental question in data analysis, machine learning and signal processing is how to compare between data points. The choice of the distance metric is specifically challenging for high-dimensional data sets, where the problem of meaningfulness is more prominent (e.g. the Euclidean distance between images). In this paper, we propose to exploit a property of high-dimensional data that is usually ignored - which is the structure stemming from the relationships between the coordinates. Specifically we show that organizing similar coordinates in clusters can be exploited for the construction of the Mahalanobis distance between samples. When the observable samples are generated by a nonlinear transformation of hidden variables, the Mahalanobis distance allows the recovery of the Euclidean distances in the hidden space.We illustrate the advantage of our approach on a synthetic example where the discovery of clusters of correlated coordinates improves the estimation of the principal directions of the samples. Our method was applied to real data of gene expression for lung adenocarcinomas (lung cancer). By using the proposed metric we found a partition of subjects to risk groups with a good separation between their Kaplan-Meier survival plot.

Ori Katz, Ronen Talmon, Yu-Lun Lo et al.

The problem of information fusion from multiple data-sets acquired by multimodal sensors has drawn significant research attention over the years. In this paper, we focus on a particular problem setting consisting of a physical phenomenon or a system of interest observed by multiple sensors. We assume that all sensors measure some aspects of the system of interest with additional sensor-specific and irrelevant components. Our goal is to recover the variables relevant to the observed system and to filter out the nuisance effects of the sensor-specific variables. We propose an approach based on manifold learning, which is particularly suitable for problems with multiple modalities, since it aims to capture the intrinsic structure of the data and relies on minimal prior model knowledge. Specifically, we propose a nonlinear filtering scheme, which extracts the hidden sources of variability captured by two or more sensors, that are independent of the sensor-specific components. In addition to presenting a theoretical analysis, we demonstrate our technique on real measured data for the purpose of sleep stage assessment based on multiple, multimodal sensor measurements. We show that without prior knowledge on the different modalities and on the measured system, our method gives rise to a data-driven representation that is well correlated with the underlying sleep process and is robust to noise and sensor-specific effects.

Vardan Papyan, Ronen Talmon

Consider a set of multiple, multimodal sensors capturing a complex system or a physical phenomenon of interest. Our primary goal is to distinguish the underlying sources of variability manifested in the measured data. The first step in our analysis is to find the common source of variability present in all sensor measurements. We base our work on a recent paper, which tackles this problem with alternating diffusion (AD). In this work, we suggest to further the analysis by extracting the sensor-specific variables in addition to the common source. We propose an algorithm, which we analyze theoretically, and then demonstrate on three different applications: a synthetic example, a toy problem, and the task of fetal ECG extraction.

Bracha Laufer-Goldshtein, Ronen Talmon, Sharon Gannot

The problem of source localization with ad hoc microphone networks in noisy and reverberant enclosures, given a training set of prerecorded measurements, is addressed in this paper. The training set is assumed to consist of a limited number of labelled measurements, attached with corresponding positions, and a larger amount of unlabelled measurements from unknown locations. However, microphone calibration is not required. We use a Bayesian inference approach for estimating a function that maps measurement-based feature vectors to the corresponding positions. The central issue is how to combine the information provided by the different microphones in a unified statistical framework. To address this challenge, we model this function using a Gaussian process with a covariance function that encapsulates both the connections between pairs of microphones and the relations among the samples in the training set. The parameters of the process are estimated by optimizing a maximum likelihood (ML) criterion. In addition, a recursive adaptation mechanism is derived where the new streaming measurements are used to update the model. Performance is demonstrated for 2-D localization of both simulated data and real-life recordings in a variety of reverberation and noise levels.

Or Yair, Ronen Talmon

In this paper, we address the problem of hidden common variables discovery from multimodal data sets of nonlinear high-dimensional observations. We present a metric based on local applications of canonical correlation analysis (CCA) and incorporate it in a kernel-based manifold learning technique.We show that this metric discovers the hidden common variables underlying the multimodal observations by estimating the Euclidean distance between them. Our approach can be viewed both as an extension of CCA to a nonlinear setting as well as an extension of manifold learning to multiple data sets. Experimental results show that our method indeed discovers the common variables underlying high-dimensional nonlinear observations without assuming prior rigid model assumptions.

David Dov, Ronen Talmon, Israel Cohen

In this paper, we address the problem of multiple view data fusion in the presence of noise and interferences. Recent studies have approached this problem using kernel methods, by relying particularly on a product of kernels constructed separately for each view. From a graph theory point of view, we analyze this fusion approach in a discrete setting. More specifically, based on a statistical model for the connectivity between data points, we propose an algorithm for the selection of the kernel bandwidth, a parameter, which, as we show, has important implications on the robustness of this fusion approach to interferences. Then, we consider the fusion of audio-visual speech signals measured by a single microphone and by a video camera pointed to the face of the speaker. Specifically, we address the task of voice activity detection, i.e., the detection of speech and non-speech segments, in the presence of structured interferences such as keyboard taps and office noise. We propose an algorithm for voice activity detection based on the audio-visual signal. Simulation results show that the proposed algorithm outperforms competing fusion and voice activity detection approaches. In addition, we demonstrate that a proper selection of the kernel bandwidth indeed leads to improved performance.

Ronen Talmon, Hau-tieng Wu

The analysis of data sets arising from multiple sensors has drawn significant research attention over the years. Traditional methods, including kernel-based methods, are typically incapable of capturing nonlinear geometric structures. We introduce a latent common manifold model underlying multiple sensor observations for the purpose of multimodal data fusion. A method based on alternating diffusion is presented and analyzed; we provide theoretical analysis of the method under the latent common manifold model. To exemplify the power of the proposed framework, experimental results in several applications are reported.

Gal Mishne, Ronen Talmon, Ron Meir et al.

In the wake of recent advances in experimental methods in neuroscience, the ability to record in-vivo neuronal activity from awake animals has become feasible. The availability of such rich and detailed physiological measurements calls for the development of advanced data analysis tools, as commonly used techniques do not suffice to capture the spatio-temporal network complexity. In this paper, we propose a new hierarchical coupled geometry analysis, which exploits the hidden connectivity structures between neurons and the dynamic patterns at multiple time-scales. Our approach gives rise to the joint organization of neurons and dynamic patterns in data-driven hierarchical data structures. These structures provide local to global data representations, from local partitioning of the data in flexible trees through a new multiscale metric to a global manifold embedding. The application of our techniques to in-vivo neuronal recordings demonstrate the capability of extracting neuronal activity patterns and identifying temporal trends, associated with particular behavioral events and manipulations introduced in the experiments.

Bracha Laufer-Goldshtein, Ronen Talmon, Sharon Gannot

Conventional speaker localization algorithms, based merely on the received microphone signals, are often sensitive to adverse conditions, such as: high reverberation or low signal to noise ratio (SNR). In some scenarios, e.g. in meeting rooms or cars, it can be assumed that the source position is confined to a predefined area, and the acoustic parameters of the environment are approximately fixed. Such scenarios give rise to the assumption that the acoustic samples from the region of interest have a distinct geometrical structure. In this paper, we show that the high dimensional acoustic samples indeed lie on a low dimensional manifold and can be embedded into a low dimensional space. Motivated by this result, we propose a semi-supervised source localization algorithm which recovers the inverse mapping between the acoustic samples and their corresponding locations. The idea is to use an optimization framework based on manifold regularization, that involves smoothness constraints of possible solutions with respect to the manifold. The proposed algorithm, termed Manifold Regularization for Localization (MRL), is implemented in an adaptive manner. The initialization is conducted with only few labelled samples attached with their respective source locations, and then the system is gradually adapted as new unlabelled samples (with unknown source locations) are received. Experimental results show superior localization performance when compared with a recently presented algorithm based on a manifold learning approach and with the generalized cross-correlation (GCC) algorithm as a baseline.