Ariel Jaffe

Ariel Jaffe, Roi Weiss, Boaz Nadler

Real eigenpairs of symmetric tensors play an important role in multiple applications. In this paper we propose and analyze a fast iterative Newton-based method to compute real eigenpairs of symmetric tensors. We derive sufficient conditions for a real eigenpair to be a stable fixed point for our method, and prove that given a sufficiently close initial guess, the convergence rate is quadratic. Empirically, our method converges to a significantly larger number of eigenpairs compared to previously proposed iterative methods, and with enough random initializations typically finds all real eigenpairs. In particular, for a generic symmetric tensor, the sufficient conditions for local convergence of our Newton-based method hold simultaneously for all its real eigenpairs.

Junchen Yang, Ofir Lindenbaum, Yuval Kluger et al.

Multi-modal high throughput biological data presents a great scientific opportunity and a significant computational challenge. In multi-modal measurements, every sample is observed simultaneously by two or more sets of sensors. In such settings, many observed variables in both modalities are often nuisance and do not carry information about the phenomenon of interest. Here, we propose a multi-modal unsupervised feature selection framework: identifying informative variables based on coupled high-dimensional measurements. Our method is designed to identify features associated with two types of latent low-dimensional structures: (i) shared structures that govern the observations in both modalities and (ii) differential structures that appear in only one modality. To that end, we propose two Laplacian-based scoring operators. We incorporate the scores with differentiable gates that mask nuisance features and enhance the accuracy of the structure captured by the graph Laplacian. The performance of the new scheme is illustrated using synthetic and real datasets, including an extended biological application to single-cell multi-omics.

Ben Leshem, Oren Raz, Ariel Jaffe et al.

In this paper we consider the following real-valued and finite dimensional specific instance of the 1-D classical phase retrieval problem. Let ${\bf F}\in\mathbb{R}^N$ be an $N$-dimensional vector, whose discrete Fourier transform has a compact support. The sign problem is to recover ${\bf F}$ from its magnitude $|{\bf F}|$. First, in contrast to the classical 1-D phase problem which in general has multiple solutions, we prove that with sufficient over-sampling, the sign problem admits a unique solution. Next, we show that the sign problem can be viewed as a special case of a more general piecewise constant phase problem. Relying on this result, we derive a computationally efficient and robust to noise sign recovery algorithm. In the noise-free case and with a sufficiently high sampling rate, our algorithm is guaranteed to recover the true sign pattern. Finally, we present two phase retrieval applications of the sign problem: (i) vectorial phase retrieval with three measurement vectors; and (ii) recovery of two well separated 1-D objects.

Ram Dyuthi Sristi, Gal Mishne, Ariel Jaffe

Selecting subsets of features that differentiate between two conditions is a key task in a broad range of scientific domains. In many applications, the features of interest form clusters with similar effects on the data at hand. To recover such clusters we develop DiSC, a data-driven approach for detecting groups of features that differentiate between conditions. For each condition, we construct a graph whose nodes correspond to the features and whose weights are functions of the similarity between them for that condition. We then apply a spectral approach to compute subsets of nodes whose connectivity differs significantly between the condition-specific feature graphs. On the theoretical front, we analyze our approach with a toy example based on the stochastic block model. We evaluate DiSC on a variety of datasets, including MNIST, hyperspectral imaging, simulated scRNA-seq and task fMRI, and demonstrate that DiSC uncovers features that better differentiate between conditions compared to competing methods.

Ortal Reshef, Ofer Glassman, Or Zuk et al.

Recovering a tree that represents the evolutionary history of a group of species is a key task in phylogenetics. Performing this task using sequence data from multiple genetic markers poses two key challenges. The first is the discordance between the evolutionary history of individual genes and that of the species. The second challenge is computational, as contemporary studies involve thousands of species. Here we present SDSR, a scalable divide-and-conquer approach for species tree reconstruction based on spectral graph theory. The algorithm recursively partitions the species into subsets until their sizes are below a given threshold. The trees of these subsets are reconstructed by a user-chosen species tree algorithm. Finally, these subtrees are merged to form the full tree. On the theoretical front, we derive recovery guarantees for SDSR, under the multispecies coalescent (MSC) model. We also perform a runtime complexity analysis. We show that SDSR, when combined with a species tree reconstruction algorithm as a subroutine, yields substantial runtime savings as compared to applying the same algorithm on the full data. Empirically, we evaluate SDSR on synthetic benchmark datasets with incomplete lineage sorting and horizontal gene transfer. In accordance with our theoretical analysis, the simulations show that combining SDSR with common species tree methods, such as CA-ML or ASTRAL, yields up to 10-fold faster runtimes. In addition, SDSR achieves a comparable tree reconstruction accuracy to that obtained by applying these methods on the full data.

Daniel Segal, Ofir Lindenbaum, Ariel Jaffe

Choosing a meaningful subset of features from high-dimensional observations in unsupervised settings can greatly enhance the accuracy of downstream analysis, such as clustering or dimensionality reduction, and provide valuable insights into the sources of heterogeneity in a given dataset. In this paper, we propose a self-supervised graph-based approach for unsupervised feature selection. Our method's core involves computing robust pseudo-labels by applying simple processing steps to the graph Laplacian's eigenvectors. The subset of eigenvectors used for computing pseudo-labels is chosen based on a model stability criterion. We then measure the importance of each feature by training a surrogate model to predict the pseudo-labels from the observations. Our approach is shown to be robust to challenging scenarios, such as the presence of outliers and complex substructures. We demonstrate the effectiveness of our method through experiments on real-world datasets, showing its robustness across multiple domains, particularly its effectiveness on biological datasets.

Yariv Aizenbud, Ariel Jaffe, Meng Wang et al.

Modeling the distribution of high dimensional data by a latent tree graphical model is a prevalent approach in multiple scientific domains. A common task is to infer the underlying tree structure, given only observations of its terminal nodes. Many algorithms for tree recovery are computationally intensive, which limits their applicability to trees of moderate size. For large trees, a common approach, termed divide-and-conquer, is to recover the tree structure in two steps. First, recover the structure separately of multiple, possibly random subsets of the terminal nodes. Second, merge the resulting subtrees to form a full tree. Here, we develop Spectral Top-Down Recovery (STDR), a deterministic divide-and-conquer approach to infer large latent tree models. Unlike previous methods, STDR partitions the terminal nodes in a non random way, based on the Fiedler vector of a suitable Laplacian matrix related to the observed nodes. We prove that under certain conditions, this partitioning is consistent with the tree structure. This, in turn, leads to a significantly simpler merging procedure of the small subtrees. We prove that STDR is statistically consistent and bound the number of samples required to accurately recover the tree with high probability. Using simulated data from several common tree models in phylogenetics, we demonstrate that STDR has a significant advantage in terms of runtime, with improved or similar accuracy.

Ariel Jaffe, Noah Amsel, Yariv Aizenbud et al.

A common assumption in multiple scientific applications is that the distribution of observed data can be modeled by a latent tree graphical model. An important example is phylogenetics, where the tree models the evolutionary lineages of a set of observed organisms. Given a set of independent realizations of the random variables at the leaves of the tree, a key challenge is to infer the underlying tree topology. In this work we develop Spectral Neighbor Joining (SNJ), a novel method to recover the structure of latent tree graphical models. Given a matrix that contains a measure of similarity between all pairs of observed variables, SNJ computes a spectral measure of cohesion between groups of observed variables. We prove that SNJ is consistent, and derive a sufficient condition for correct tree recovery from an estimated similarity matrix. Combining this condition with a concentration of measure result on the similarity matrix, we bound the number of samples required to recover the tree with high probability. We illustrate via extensive simulations that in comparison to several other reconstruction methods, SNJ requires fewer samples to accurately recover trees with a large number of leaves or long edges.

Ariel Jaffe, Yuval Kluger, Ofir Lindenbaum et al.

word2vec due to Mikolov \textit{et al.} (2013) is a word embedding method that is widely used in natural language processing. Despite its great success and frequent use, theoretical justification is still lacking. The main contribution of our paper is to propose a rigorous analysis of the highly nonlinear functional of word2vec. Our results suggest that word2vec may be primarily driven by an underlying spectral method. This insight may open the door to obtaining provable guarantees for word2vec. We support these findings by numerical simulations. One fascinating open question is whether the nonlinear properties of word2vec that are not captured by the spectral method are beneficial and, if so, by what mechanism.

Ariel Jaffe, Roi Weiss, Shai Carmi et al.

Latent variable models with hidden binary units appear in various applications. Learning such models, in particular in the presence of noise, is a challenging computational problem. In this paper we propose a novel spectral approach to this problem, based on the eigenvectors of both the second order moment matrix and third order moment tensor of the observed data. We prove that under mild non-degeneracy conditions, our method consistently estimates the model parameters at the optimal parametric rate. Our tensor-based method generalizes previous orthogonal tensor decomposition approaches, where the hidden units were assumed to be either statistically independent or mutually exclusive. We illustrate the consistency of our method on simulated data and demonstrate its usefulness in learning a common model for population mixtures in genetics.

Amit Moscovich, Ariel Jaffe, Boaz Nadler

We consider semi-supervised regression when the predictor variables are drawn from an unknown manifold. A simple two step approach to this problem is to: (i) estimate the manifold geodesic distance between any pair of points using both the labeled and unlabeled instances; and (ii) apply a k nearest neighbor regressor based on these distance estimates. We prove that given sufficiently many unlabeled points, this simple method of geodesic kNN regression achieves the optimal finite-sample minimax bound on the mean squared error, as if the manifold were known. Furthermore, we show how this approach can be efficiently implemented, requiring only O(k N log N) operations to estimate the regression function at all N labeled and unlabeled points. We illustrate this approach on two datasets with a manifold structure: indoor localization using WiFi fingerprints and facial pose estimation. In both cases, geodesic kNN is more accurate and much faster than the popular Laplacian eigenvector regressor.

Uri Shaham, Xiuyuan Cheng, Omer Dror et al.

We show how deep learning methods can be applied in the context of crowdsourcing and unsupervised ensemble learning. First, we prove that the popular model of Dawid and Skene, which assumes that all classifiers are conditionally independent, is {\em equivalent} to a Restricted Boltzmann Machine (RBM) with a single hidden node. Hence, under this model, the posterior probabilities of the true labels can be instead estimated via a trained RBM. Next, to address the more general case, where classifiers may strongly violate the conditional independence assumption, we propose to apply RBM-based Deep Neural Net (DNN). Experimental results on various simulated and real-world datasets demonstrate that our proposed DNN approach outperforms other state-of-the-art methods, in particular when the data violates the conditional independence assumption.

Ariel Jaffe, Ethan Fetaya, Boaz Nadler et al.

In unsupervised ensemble learning, one obtains predictions from multiple sources or classifiers, yet without knowing the reliability and expertise of each source, and with no labeled data to assess it. The task is to combine these possibly conflicting predictions into an accurate meta-learner. Most works to date assumed perfect diversity between the different sources, a property known as conditional independence. In realistic scenarios, however, this assumption is often violated, and ensemble learners based on it can be severely sub-optimal. The key challenges we address in this paper are:\ (i) how to detect, in an unsupervised manner, strong violations of conditional independence; and (ii) construct a suitable meta-learner. To this end we introduce a statistical model that allows for dependencies between classifiers. Our main contributions are the development of novel unsupervised methods to detect strongly dependent classifiers, better estimate their accuracies, and construct an improved meta-learner. Using both artificial and real datasets, we showcase the importance of taking classifier dependencies into account and the competitive performance of our approach.

Ariel Jaffe, Boaz Nadler, Yuval Kluger

In various situations one is given only the predictions of multiple classifiers over a large unlabeled test data. This scenario raises the following questions: Without any labeled data and without any a-priori knowledge about the reliability of these different classifiers, is it possible to consistently and computationally efficiently estimate their accuracies? Furthermore, also in a completely unsupervised manner, can one construct a more accurate unsupervised ensemble classifier? In this paper, focusing on the binary case, we present simple, computationally efficient algorithms to solve these questions. Furthermore, under standard classifier independence assumptions, we prove our methods are consistent and study their asymptotic error. Our approach is spectral, based on the fact that the off-diagonal entries of the classifiers' covariance matrix and 3-d tensor are rank-one. We illustrate the competitive performance of our algorithms via extensive experiments on both artificial and real datasets.