arXiv subjects

Robert Nowak

Publications and source records attributed to Robert Nowak.

At least 91 records · Page 5Linked to original sources

arXiv2014

Data Requirement for Phylogenetic Inference from Multiple Loci: A New Distance Method

We consider the problem of estimating the evolutionary history of a set of species (phylogeny or species tree) from several genes. It is known that the evolutionary history of individual genes (gene trees) might be topologically distinct from each other and from the underlying species tree, possibly confounding phylogenetic analysis. A further complication in practice is that one has to estimate gene trees from molecular sequences of finite length. We provide the first full data-requirement analysis of a species tree reconstruction method that takes into account estimation errors at the gene level. Under that criterion, we also devise a novel reconstruction algorithm that provably improves over all previous methods in a regime of interest.

q-bio.PE↗

arXiv2013

lil' UCB : An Optimal Exploration Algorithm for Multi-Armed Bandits

The paper proposes a novel upper confidence bound (UCB) procedure for identifying the arm with the largest mean in a multi-armed bandit game in the fixed confidence setting using a small number of total samples. The procedure cannot be improved in the sense that the number of samples required to identify the best arm is within a constant factor of a lower bound based on the law of the iterated logarithm (LIL). Inspired by the LIL, we construct our confidence bounds to explicitly account for the infinite time horizon of the algorithm. In addition, by using a novel stopping time for the algorithm we avoid a union bound over the arms that has been observed in other UCB-type algorithms. We prove that the algorithm is optimal up to constants and also show through simulations that it provides superior performance with respect to the state-of-the-art.

stat.ML↗

arXiv2013

A Junction Tree Framework for Undirected Graphical Model Selection

An undirected graphical model is a joint probability distribution defined on an undirected graph G*, where the vertices in the graph index a collection of random variables and the edges encode conditional independence relationships among random variables. The undirected graphical model selection (UGMS) problem is to estimate the graph G* given observations drawn from the undirected graphical model. This paper proposes a framework for decomposing the UGMS problem into multiple subproblems over clusters and subsets of the separators in a junction tree. The junction tree is constructed using a graph that contains a superset of the edges in G*. We highlight three main properties of using junction trees for UGMS. First, different regularization parameters or different UGMS algorithms can be used to learn different parts of the graph. This is possible since the subproblems we identify can be solved independently of each other. Second, under certain conditions, a junction tree based UGMS algorithm can produce consistent results with fewer observations than the usual requirements of existing algorithms. Third, both our theoretical and experimental results show that the junction tree framework does a significantly better job at finding the weakest edges in a graph than existing methods. This property is a consequence of both the first and second properties. Finally, we note that our framework is independent of the choice of the UGMS algorithm and can be used as a wrapper around standard UGMS algorithms for more accurate graph estimation.

stat.ML↗

arXiv2013

Sparse Overlapping Sets Lasso for Multitask Learning and its Application to fMRI Analysis

Multitask learning can be effective when features useful in one task are also useful for other tasks, and the group lasso is a standard method for selecting a common subset of features. In this paper, we are interested in a less restrictive form of multitask learning, wherein (1) the available features can be organized into subsets according to a notion of similarity and (2) features useful in one task are similar, but not necessarily identical, to the features best suited for other tasks. The main contribution of this paper is a new procedure called Sparse Overlapping Sets (SOS) lasso, a convex optimization that automatically selects similar features for related learning tasks. Error bounds are derived for SOSlasso and its consistency is established for squared error loss. In particular, SOSlasso is motivated by multi- subject fMRI studies in which functional activity is classified using brain voxels as features. Experiments with real and synthetic data demonstrate the advantages of SOSlasso compared to the lasso and group lasso.

cs.LG↗

arXiv2013

On Finding the Largest Mean Among Many

Sampling from distributions to find the one with the largest mean arises in a broad range of applications, and it can be mathematically modeled as a multi-armed bandit problem in which each distribution is associated with an arm. This paper studies the sample complexity of identifying the best arm (largest mean) in a multi-armed bandit problem. Motivated by large-scale applications, we are especially interested in identifying situations where the total number of samples that are necessary and sufficient to find the best arm scale linearly with the number of arms. We present a single-parameter multi-armed bandit model that spans the range from linear to superlinear sample complexity. We also give a new algorithm for best arm identification, called PRISM, with linear sample complexity for a wide range of mean distributions. The algorithm, like most exploration procedures for multi-armed bandits, is adaptive in the sense that the next arms to sample are selected based on previous samples. We compare the sample complexity of adaptive procedures with simpler non-adaptive procedures using new lower bounds. For many problem instances, the increased sample complexity required by non-adaptive procedures is a polynomial factor of the number of arms.

stat.ML↗

arXiv2013

Sketching Sparse Matrices

This paper considers the problem of recovering an unknown sparse p\times p matrix X from an m\times m matrix Y=AXB^T, where A and B are known m \times p matrices with m << p. The main result shows that there exist constructions of the "sketching" matrices A and B so that even if X has O(p) non-zeros, it can be recovered exactly and efficiently using a convex program as long as these non-zeros are not concentrated in any single row/column of X. Furthermore, it suffices for the size of Y (the sketch dimension) to scale as m = O(\sqrt{# nonzeros in X} \times log p). The results also show that the recovery is robust and stable in the sense that if X is equal to a sparse matrix plus a perturbation, then the convex program we propose produces an approximation with accuracy proportional to the size of the perturbation. Unlike traditional results on sparse recovery, where the sensing matrix produces independent measurements, our sensing operator is highly constrained (it assumes a tensor product structure). Therefore, proving recovery guarantees require non-standard techniques. Indeed our approach relies on a novel result concerning tensor products of bipartite graphs, which may be of independent interest. This problem is motivated by the following application, among others. Consider a p\times n data matrix D, consisting of n observations of p variables. Assume that the correlation matrix X:=DD^{T} is (approximately) sparse in the sense that each of the p variables is significantly correlated with only a few others. Our results show that these significant correlations can be detected even if we have access to only a sketch of the data S=AD with A \in R^{m\times p}.

cs.IT↗

arXiv2012

Signal Recovery in Unions of Subspaces with Applications to Compressive Imaging

In applications ranging from communications to genetics, signals can be modeled as lying in a union of subspaces. Under this model, signal coefficients that lie in certain subspaces are active or inactive together. The potential subspaces are known in advance, but the particular set of subspaces that are active (i.e., in the signal support) must be learned from measurements. We show that exploiting knowledge of subspaces can further reduce the number of measurements required for exact signal recovery, and derive universal bounds for the number of measurements needed. The bound is universal in the sense that it only depends on the number of subspaces under consideration, and their orientation relative to each other. The particulars of the subspaces (e.g., compositions, dimensions, extents, overlaps, etc.) does not affect the results we obtain. In the process, we derive sample complexity bounds for the special case of the group lasso with overlapping groups (the latent group lasso), which is used in a variety of applications. Finally, we also show that wavelet transform coefficients of images can be modeled as lying in groups, and hence can be efficiently recovered using group lasso methods.

stat.ML↗

arXiv2012

Robust Spatio-Temporal Signal Recovery from Noisy Counts in Social Media

Many real-world phenomena can be represented by a spatio-temporal signal: where, when, and how much. Social media is a tantalizing data source for those who wish to monitor such signals. Unlike most prior work, we assume that the target phenomenon is known and we are given a method to count its occurrences in social media. However, counting is plagued by sample bias, incomplete data, and, paradoxically, data scarcity -- issues inadequately addressed by prior work. We formulate signal recovery as a Poisson point process estimation problem. We explicitly incorporate human population bias, time delays and spatial distortions, and spatio-temporal regularization into the model to address the noisy count issues. We present an efficient optimization algorithm and discuss its theoretical properties. We show that our model is more accurate than commonly-used baselines. Finally, we present a case study on wildlife roadkill monitoring, where our model produces qualitatively convincing results.

cs.AI↗

arXiv2012

Minimax-Optimal Bounds for Detectors Based on Estimated Prior Probabilities

In many signal detection and classification problems, we have knowledge of the distribution under each hypothesis, but not the prior probabilities. This paper is aimed at providing theory to quantify the performance of detection via estimating prior probabilities from either labeled or unlabeled training data. The error or {\em risk} is considered as a function of the prior probabilities. We show that the risk function is locally Lipschitz in the vicinity of the true prior probabilities, and the error of detectors based on estimated prior probabilities depends on the behavior of the risk function in this locality. In general, we show that the error of detectors based on the Maximum Likelihood Estimate (MLE) of the prior probabilities converges to the Bayes error at a rate of $n^{-1/2}$, where $n$ is the number of training data. If the behavior of the risk function is more favorable, then detectors based on the MLE have errors converging to the corresponding Bayes errors at optimal rates of the form $n^{-(1+α)/2}$, where $α>0$ is a parameter governing the behavior of the risk function with a typical value $α= 1$. The limit $α\rightarrow \infty$ corresponds to a situation where the risk function is flat near the true probabilities, and thus insensitive to small errors in the MLE; in this case the error of the detector based on the MLE converges to the Bayes error exponentially fast with $n$. We show the bounds are achievable no matter given labeled or unlabeled training data and are minimax-optimal in labeled case.

cs.IT↗

arXiv2011

High-Rank Matrix Completion and Subspace Clustering with Missing Data

This paper considers the problem of completing a matrix with many missing entries under the assumption that the columns of the matrix belong to a union of multiple low-rank subspaces. This generalizes the standard low-rank matrix completion problem to situations in which the matrix rank can be quite high or even full rank. Since the columns belong to a union of subspaces, this problem may also be viewed as a missing-data version of the subspace clustering problem. Let X be an n x N matrix whose (complete) columns lie in a union of at most k subspaces, each of rank <= r < n, and assume N >> kn. The main result of the paper shows that under mild assumptions each column of X can be perfectly recovered with high probability from an incomplete version so long as at least CrNlog^2(n) entries of X are observed uniformly at random, with C>1 a constant depending on the usual incoherence conditions, the geometrical arrangement of subspaces, and the distribution of columns over the subspaces. The result is illustrated with numerical experiments and an application to Internet distance matrix completion and topology identification.

cs.IT↗

arXiv2011

On the Limits of Sequential Testing in High Dimensions

This paper presents results pertaining to sequential methods for support recovery of sparse signals in noise. Specifically, we show that any sequential measurement procedure fails provided the average number of measurements per dimension grows slower then log s / D(f0||f1) where s is the level of sparsity, and D(f0||f1) the Kullback-Leibler divergence between the underlying distributions. For comparison, we show any non-sequential procedure fails provided the number of measurements grows at a rate less than log n / D(f1||f0), where n is the total dimension of the problem. Lastly, we show that a simple procedure termed sequential thresholding guarantees exact support recovery provided the average number of measurements per dimension grows faster than (log s + log log n) / D(f0||f1), a mere additive factor more than the lower bound.

cs.IT↗

arXiv2011

Tight Measurement Bounds for Exact Recovery of Structured Sparse Signals

Standard compressive sensing results state that to exactly recover an s sparse signal in R^p, one requires O(s. log(p)) measurements. While this bound is extremely useful in practice, often real world signals are not only sparse, but also exhibit structure in the sparsity pattern. We focus on group-structured patterns in this paper. Under this model, groups of signal coefficients are active (or inactive) together. The groups are predefined, but the particular set of groups that are active (i.e., in the signal support) must be learned from measurements. We show that exploiting knowledge of groups can further reduce the number of measurements required for exact signal recovery, and derive universal bounds for the number of measurements needed. The bound is universal in the sense that it only depends on the number of groups under consideration, and not the particulars of the groups (e.g., compositions, sizes, extents, overlaps, etc.). Experiments show that our result holds for a variety of overlapping group configurations.

stat.ML↗

arXiv2011

Online Identification and Tracking of Subspaces from Highly Incomplete Information

This work presents GROUSE (Grassmanian Rank-One Update Subspace Estimation), an efficient online algorithm for tracking subspaces from highly incomplete observations. GROUSE requires only basic linear algebraic manipulations at each iteration, and each subspace update can be performed in linear time in the dimension of the subspace. The algorithm is derived by analyzing incremental gradient descent on the Grassmannian manifold of subspaces. With a slight modification, GROUSE can also be used as an online incremental algorithm for the matrix completion problem of imputing missing entries of a low-rank matrix. GROUSE performs exceptionally well in practice both in tracking subspaces and as an online algorithm for matrix completion.

cs.IT↗

arXiv2011

Sequential Analysis in High Dimensional Multiple Testing and Sparse Recovery

This paper studies the problem of high-dimensional multiple testing and sparse recovery from the perspective of sequential analysis. In this setting, the probability of error is a function of the dimension of the problem. A simple sequential testing procedure is proposed. We derive necessary conditions for reliable recovery in the non-sequential setting and contrast them with sufficient conditions for reliable recovery using the proposed sequential testing procedure. Applications of the main results to several commonly encountered models show that sequential testing can be exponentially more sensitive to the difference between the null and alternative distributions (in terms of the dependence on dimension), implying that subtle cases can be much more reliably determined using sequential methods.

math.ST↗

arXiv2011

Causal Network Inference via Group Sparse Regularization

This paper addresses the problem of inferring sparse causal networks modeled by multivariate auto-regressive (MAR) processes. Conditions are derived under which the Group Lasso (gLasso) procedure consistently estimates sparse network structure. The key condition involves a "false connection score." In particular, we show that consistent recovery is possible even when the number of observations of the network is far less than the number of parameters describing the network, provided that the false connection score is less than one. The false connection score is also demonstrated to be a useful metric of recovery in non-asymptotic regimes. The conditions suggest a modified gLasso procedure which tends to improve the false connection score and reduce the chances of reversing the direction of causal influence. Computational experiments and a real network based electrocorticogram (ECoG) simulation study demonstrate the effectiveness of the approach.

stat.ML↗

arXiv2011

Active Clustering: Robust and Efficient Hierarchical Clustering using Adaptively Selected Similarities

Hierarchical clustering based on pairwise similarities is a common tool used in a broad range of scientific applications. However, in many problems it may be expensive to obtain or compute similarities between the items to be clustered. This paper investigates the hierarchical clustering of N items based on a small subset of pairwise similarities, significantly less than the complete set of N(N-1)/2 similarities. First, we show that if the intracluster similarities exceed intercluster similarities, then it is possible to correctly determine the hierarchical clustering from as few as 3N log N similarities. We demonstrate this order of magnitude savings in the number of pairwise similarities necessitates sequentially selecting which similarities to obtain in an adaptive fashion, rather than picking them at random. We then propose an active clustering method that is robust to a limited fraction of anomalous similarities, and show how even in the presence of these noisy similarity values we can resolve the hierarchical clustering using only O(N log^2 N) pairwise similarities.

cs.IT↗

arXiv2011

High-Dimensional Matched Subspace Detection When Data are Missing

We consider the problem of deciding whether a highly incomplete signal lies within a given subspace. This problem, Matched Subspace Detection, is a classical, well-studied problem when the signal is completely observed. High- dimensional testing problems in which it may be prohibitive or impossible to obtain a complete observation motivate this work. The signal is represented as a vector in R^n, but we only observe m << n of its elements. We show that reliable detection is possible, under mild incoherence conditions, as long as m is slightly greater than the dimension of the subspace in question.

cs.IT↗

arXiv2010

Distilled Sensing: Adaptive Sampling for Sparse Detection and Estimation

Adaptive sampling results in dramatic improvements in the recovery of sparse signals in white Gaussian noise. A sequential adaptive sampling-and-refinement procedure called Distilled Sensing (DS) is proposed and analyzed. DS is a form of multi-stage experimental design and testing. Because of the adaptive nature of the data collection, DS can detect and localize far weaker signals than possible from non-adaptive measurements. In particular, reliable detection and localization (support estimation) using non-adaptive samples is possible only if the signal amplitudes grow logarithmically with the problem dimension. Here it is shown that using adaptive sampling, reliable detection is possible provided the amplitude exceeds a constant, and localization is possible when the amplitude exceeds any arbitrarily slowly growing function of the dimension.

math.ST↗