Yu Xia, Subhojyoti Mukherjee, Zhouhang Xie, Junda Wu, Xintong Li, Ryan Aponte, Hanjia Lyu, Joe Barrow, Hongjie Chen, Franck Dernoncourt, Branislav Kveton, Tong Yu, Ruiyi Zhang, Jiuxiang Gu, Nesreen K. Ahmed, Yu Wang, Xiang Chen, Hanieh Deilamsalehy, Sungchul Kim, Zhengmian Hu, Yue Zhao, Nedim Lipka, Seunghyun Yoon, Ting-Hao Kenneth Huang, Zichao Wang, Puneet Mathur, Soumyabrata Pal, Koyel Mukherjee, Zhehao Zhang, Namyong Park, Thien Huu Nguyen, Jiebo Luo, Ryan A. Rossi, Julian McAuley. Proceedings of the 63rd Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers). 2025.

Soumya Suvra Ghosal, Soumyabrata Pal, Koyel Mukherjee, Dinesh Manocha. Proceedings of the 2025 Conference of the Nations of the Americas Chapter of the Association for Computational Linguistics: Human Language Technologies (Volume 1: Long Papers). 2025.

Mixture models are widely used to fit complex and multimodal datasets. In this paper we study mixtures with high dimensional sparse parameter vectors and consider the problem of support recovery of those vectors. While parameter learning in mixture models is well-studied, the sparsity constraint remains relatively unexplored. Sparsity of parameter vectors is a natural assumption in high dimensional settings, and support recovery is a major step towards parameter estimation. We provide efficient algorithms for support recovery that have a logarithmic sample complexity dependence on the dimensionality of the latent space, and also poly-logarithmic dependence on sparsity. Our algorithms, applicable to mixtures of many different canonical distributions including high dimensional Uniform, Poisson, Laplace, Gaussians, etc., are based on themethod of moments. In most of these settings, our results are the first guarantees on the problem while in the rest, our results provide improvements on or are competitive with existing works.

Personalization of machine learning (ML) predictions for individual users/domains/enterprises is critical for practical recommendation systems. Standard personalization approaches involve learning a user/domain specific \emph{embedding} that is fed into a fixed global model which can be limiting. On the other hand, personalizing/fine-tuning model itself for each user/domain — a.k.a meta-learning — has high storage/infrastructure cost. Moreover, rigorous theoretical studies of scalable personalization approaches have been very limited. To address the above issues, we propose a novel meta-learning style approach that models network weights as a sum of low-rank and sparse components. This captures common information from multiple individuals/users together in the low-rank part while sparse part captures user-specific idiosyncrasies. We then study the framework in the linear setting, where the problem reduces to that of estimating the sum of a rank-$r$ and a $k$-column sparse matrix using a small number of linear measurements. We propose a computationally efficient alternating minimization method with iterative hard thresholding — AMHT-LRS — to learn the low-rank and sparse part. Theoretically, for the realizable Gaussian data setting, we show that AMHT-LRS solves the problem efficiently with nearly optimal sample complexity. Finally, a significant challenge in personalization is ensuring privacy of each user’s sensitive data. We alleviate this problem by proposing a differentially private variant of our method that also is equipped with strong generalization guarantees.

We investigate the low rank matrix completion problem in an online setting with ${M}$ users, ${N}$ items, ${T}$ rounds, and an unknown rank-$r$ reward matrix ${R}\in \mathbb{R}^{{M}\times {N}}$. This problem has been well-studied in the literature and has several applications in practice. In each round, we recommend ${S}$ carefully chosen distinct items to every user and observe noisy rewards. In the regime where ${M},{N} >> {T}$, we propose two distinct computationally efficient algorithms for recommending items to users and analyze them under the benign *hott items* assumption 1) First, for ${S}=1$, under additional incoherence/smoothness assumptions on ${R}$, we propose the phased algorithm PhasedClusterElim. Our algorithm obtains a near-optimal per-user regret of $\tilde{O}({N}{M}^{-1}(\Delta^{-1}+\Delta_{\text{hott}}^{-2}))$ where $\Delta_{\text{hott}},\Delta$ are problem-dependent gap parameters with $\Delta_{\text{hott}} >> \Delta$ almost always. 2) Second, we consider a simplified setting with ${S}=r$ where we make significantly milder assumptions on ${R}$. Here, we introduce another phased algorithm, DeterminantElim, to derive a regret guarantee of $\tilde{O}({N}{M}^{-1/r}\Delta_\text{det}^{-1}))$ where $\Delta_{\text{det}}$ is another problem-dependent gap. Both algorithms crucially use collaboration among users to jointly eliminate sub-optimal items for groups of users successively in phases, but with distinctive and novel approaches.

$\newcommand{\Tr}{\mathsf{Tr}}$ We consider the problem of high-dimensional heavy-tailed statistical estimation in the streaming setting, which is much harder than the traditional batch setting due to memory constraints. We cast this problem as stochastic convex optimization with heavy tailed stochastic gradients, and prove that the widely used Clipped-SGD algorithm attains near-optimal sub-Gaussian statistical rates whenever the second moment of the stochastic gradient noise is finite. More precisely, with $T$ samples, we show that Clipped-SGD, for smooth and strongly convex objectives, achieves an error of $\sqrt{\frac{\Tr(\Sigma)+\sqrt{\Tr(\Sigma)\\|\Sigma\\|_2}\ln(\tfrac{\ln(T)}{\delta})}{T}}$ with probability $1-\delta$, where $\Sigma$ is the covariance of the clipped gradient. Note that the fluctuations (depending on $\tfrac{1}{\delta}$) are of lower order than the term $\Tr(\Sigma)$. This improves upon the current best rate of $\sqrt{\frac{\Tr(\Sigma)\ln(\tfrac{1}{\delta})}{T}}$ for Clipped-SGD, known \emph{only} for smooth and strongly convex objectives. Our results also extend to smooth convex and lipschitz convex objectives. Key to our result is a novel iterative refinement strategy for martingale concentration, improving upon the PAC-Bayes approach of \citet{catoni2018dimension}.

The planted densest subgraph detection problem refers to the task of testing whether in a given (random) graph there is a subgraph that is unusually dense. Specifically, we observe an undirected and unweighted graph on $n$ vertices. Under the null hypothesis, the graph is a realization of an Erdös-Rényi graph with edge probability (or, density) $q$. Under the alternative, there is a subgraph on $k$ vertices with edge probability $p>q$. The statistical as well as the computational barriers of this problem are well-understood for a wide range of the edge parameters $p$ and $q$. In this paper, we consider a natural variant of the above problem, where one can only observe a relatively small part of the graph using adaptive edge queries. For this model, we determine the number of queries necessary and sufficient (accompanied with a quasi-polynomial optimal algorithm) for detecting the presence of the planted subgraph. We also propose a polynomial-time algorithm which is able to detect the planted subgraph, albeit with more queries compared to the above lower bound. We conjecture that in the leftover regime, no polynomial-time algorithms exist. Our results resolve two open questions posed in the past literature.

One-bit compressed sensing (1bCS) is an extremely quantized signal acquisition method that has been proposed and studied rigorously in the past decade. In 1bCS, linear samples of a high dimensional signal are quantized to only one bit per sample (sign of the measurement). The extreme quantization makes it an interesting case study of the more general single-index or generalized linear models. At the same time it can also be thought of as a ‘design’ version of learning a binary linear classifier or halfspace-learning. Assuming the original signal vector to be sparse, existing results in 1bCS either aim to find the support of the vector, or approximate the signal allowing a small error. The focus of this paper is support recovery, which often also computationally facilitate approximate signal recovery. A universal measurement matrix for 1bCS refers to one set of measurements that work for all sparse signals. With universality, it is known that$\tilde {\Theta }(k^{2})~1$bCS measurements are necessary and sufficient for support recovery (where$k$denotes the sparsity). To improve the dependence on sparsity from quadratic to linear, in this work we propose approximate support recovery (allowing$\epsilon >0$proportion of errors), and superset recovery (allowing$\epsilon $proportion of false positives). We show that the first type of recovery is possible with$\tilde {O}(k/\epsilon )$measurements, while the later type of recovery, more challenging, is possible with$\tilde {O}(\max \{k/\epsilon ,k^{3/2}\}) ^{^{^{^{}}}}$measurements. We also show that in both cases$\Omega (k/\epsilon )$measurements would be necessary for universal recovery. Improved results are possible if we consider universal recovery within a restricted class of signals, such as rational signals, or signals with bounded dynamic range. In both cases superset recovery is possible with only$\tilde {O}(k/\epsilon )$measurements. Other results on universal but approximate support recovery are also provided in this paper. All of our main recovery algorithms are simple and polynomial-time.

We consider the problem of latent bandits with cluster structure where there are multiple users, each with an associated multi-armed bandit problem. These users are grouped into latent clusters such that the mean reward vectors of users within the same cluster are identical. At each round, a user, selected uniformly at random, pulls an arm and observes a corresponding noisy reward. The goal of the users is to maximize their cumulative rewards. This problem is central to practical recommendation systems and has received wide attention of late Gentile et al. (2014), Maillard and Mannor (2014). Now, if each user acts independently, then they would have to explore each arm independently and a regret of $\Omega(\sqrt{\mathrm{MNT}})$ is unavoidable, where M, N are the number of arms and users, respectively. Instead, we propose LATTICE (Latent bAndiTs via maTrIx ComplEtion) which allows exploration of the latent cluster structure to provide the minimax optimal regret of $\widetilde{O}(\sqrt{(M+N)T})$ when the number of clusters is $\tilde O(1)$. This is the first algorithm to guarantee such strong regret bound. LATTICE is based on a careful exploitation of arm information within a cluster while simultaneously clustering users. Furthermore, it is computationally efficient and requires only $O(\log \mathrm{T})$ calls to an offline matrix completion oracle across all T rounds.

We consider the problem of \emph{blocked} collaborative bandits where there are multiple users, each with an associated multi-armed bandit problem. These users are grouped into \emph{latent} clusters such that the mean reward vectors of users within the same cluster are identical. Our goal is to design algorithms that maximize the cumulative reward accrued by all the users over time, under the \emph{constraint} that no arm of a user is pulled more than $\mathsf{B}$ times. This problem has been originally considered by \cite{Bresler:2014}, and designing regret-optimal algorithms for it has since remained an open problem. In this work, we propose an algorithm called B-LATTICE (Blocked Latent bAndiTs via maTrIx ComplEtion) that collaborates across users, while simultaneously satisfying the budget constraints, to maximize their cumulative rewards. Theoretically, under certain reasonable assumptions on the latent structure, with $\mathsf{M}$ users, $\mathsf{N}$ arms, $\mathsf{T}$ rounds per user, and $\mathsf{C}=O(1)$ latent clusters, B-LATTICE achieves a per-user regret of $\widetilde{O}(\sqrt{\mathsf{T}(1 + \mathsf{N}\mathsf{M}^{-1})})$ under a budget constraint of $\mathsf{B}=\Theta(\log \mathsf{T})$. These are the first sub-linear regret bounds for this problem, and match the minimax regret bounds when $\mathsf{B}=\mathsf{T}$. Empirically, we demonstrate that our algorithm has superior performance over baselines even when $\mathsf{B}=1$. B-LATTICE is a phased algorithm where in each phase it clusters users into groups and collaborates across users within a group to quickly learn their reward models.

We obtain essentially tight upper bounds for a strengthened notion of regret in the stochastic linear bandits framework. The strengthening---referred to as Nash regret---is defined as the difference between the (a priori unknown) optimum and the geometric mean of expected rewards accumulated by the linear bandit algorithm. Since the geometric mean corresponds to the well-studied Nash social welfare (NSW) function, this formulation quantifies the performance of a bandit algorithm as the collective welfare it generates across rounds. NSW is known to satisfy fairness axioms and, hence, an upper bound on Nash regret provides a principled fairness guarantee. We consider the stochastic linear bandits problem over a horizon of $\mathsf{T}$ rounds and with a set of arms ${\cal X}$ in ambient dimension $d$. Furthermore, we focus on settings in which the stochastic reward---associated with each arm in ${\cal X}$---is a non-negative, sub-Poisson random variable. For this setting, we develop an algorithm that achieves a Nash regret of $O\left( \sqrt{\frac{d}{\mathsf{T}}} \log(\mathsf{T} |{\cal X}|)\right)$. In addition, addressing linear bandit instances in which the set of arms ${\cal X}$ is not necessarily finite, we obtain a Nash regret upper bound of $O\left( \frac{d^\frac{5}{4}}{\sqrt{\mathsf{T}}} \log(\mathsf{T})\right)$. Since bounded random variables are sub-Poisson, these results hold for bounded, non-negative rewards. Our linear bandit algorithm is built upon the successive elimination method with novel technical insights, including tailored concentration bounds and the use of sampling via John ellipsoid in conjunction with the Kiefer–Wolfowitz optimal design.

To capture the inherent geometric features of many community detection problems, we propose to use a new random graph model of communities that we call a Geometric Block Model. The geometric block model builds on the random geometric graphs (Gilbert, 1961), one of the basic models of random graphs for spatial networks, in the same way that the well-studied stochastic block model builds on the Erdos-Renyi random graphs. It is also a natural extension of random community models inspired by the recent theoretical and practical advancements in community detection. To analyze the geometric block model, we first provide new connectivity results for random annulus graphs which are generalizations of random geometric graphs. The connectivity properties of geometric graphs have been studied since their introduction, and analyzing them has been more difficult than their Erdos-Renyi counterparts, due to correlated edge formation. We then use the connectivity results of random annulus graphs to provide necessary and sufficient conditions for efficient recovery of communities for the geometric block model. We show that a simple triangle-counting algorithm to detect communities in the geometric block model is near-optimal. For this, we consider the following two regimes of graph density. In the regime where the average degree of the graph grows logarithmically with the number of vertices, we show that our algorithm performs extremely well, both theoretically and practically. In contrast, the triangle-counting algorithm is far from being optimum for the stochastic block model in the logarithmic degree regime. We simulate our results on both real and synthetic datasets to show superior performance of both the new model as well as our algorithm.

Mixture models are widely used to fit complex and multimodal datasets. In this paper we study mixtures with high dimensional sparse latent parameter vectors and consider the problem of support recovery of those vectors. While parameter learning in mixture models is well-studied, the sparsity constraint remains relatively unexplored. Sparsity of parameter vectors is a natural constraint in variety of settings, and support recovery is a major step towards parameter estimation. We provide efficient algorithms for support recovery that have a logarithmic sample complexity dependence on the dimensionality of the latent space. Our algorithms are quite general, namely they are applicable to 1) mixtures of many different canonical distributions including Uniform, Poisson, Laplace, Gaussians, etc. 2) Mixtures of linear regressions and linear classifiers with Gaussian covariates under different assumptions on the unknown parameters. In most of these settings, our results are the first guarantees on this problem while in the rest, we provide significant improvements on existing results in certain regimes.

Mixtures of high dimensional Gaussian distributions have been studied extensively in statistics and learning theory. While the total variation distance appears naturally in the sample complexity of distribution learning, it is analytically difficult to obtain tight lower bounds for mixtures. Exploiting a connection between total variation distance and the characteristic function of the mixture, we provide fairly tight functional approximations. This enables us to derive new lower bounds on the total variation distance between two-component Gaussian mixtures with a shared covariance matrix.

While mixture of linear regressions (MLR) is a well-studied topic, prior works usually do not analyze such models for prediction error. In fact, prediction and loss are not well-defined in the context of mixtures. In this paper, first we show that MLR can be used for prediction where instead of predicting a label, the model predicts a list of values (also known as list-decoding). The list size is equal to the number of components in the mixture, and the loss function is defined to be minimum among the losses resulted by all the component models. We show that with this definition, a solution of the empirical risk minimization (ERM) achieves small probability of prediction error. This begs for an algorithm to minimize the empirical risk for MLR, which is known to be computationally hard. Prior algorithmic works in MLR focus on the realizable setting, i.e., recovery of parameters when data is probabilistically generated by a mixed linear (noisy) model. In this paper we show that a version of the popular expectation minimization (EM) algorithm finds out the best fit lines in a dataset even when a realizable model is not assumed, under some regularity conditions on the dataset and the initial points, and thereby provides a solution for the ERM. We further provide an algorithm that runs in polynomial time in the number of datapoints, and recovers a good approximation of the best fit lines. The two algorithms are experimentally compared.

Recommendation systems often use online collaborative filtering (CF) algorithms to identify items a given user likes over time, based on ratings that this user and a large number of other users have provided in the past. This problem has been studied extensively when users’ preferences do not change over time (static case); an assumption that is often violated in practical settings. In this paper, we introduce a novel model for online non-stationary recommendation systems which allows for temporal uncertainties in the users’ preferences. For this model, we propose a user-based CF algorithm, and provide a theoretical analysis of its achievable reward. Compared to related non-stationary multi-armed bandit literature, the main fundamental difficulty in our model lies in the fact that variations in the preferences of a certain user may affect the recommendations for other users severely. We also test our algorithm over real-world datasets, showing its effectiveness in real-world applications. One of the main surprising observations in our experiments is the fact our algorithm outperforms other static algorithms even when preferences do not change over time. This hints toward the general conclusion that in practice, dynamic algorithms, such as the one we propose, might be beneficial even in stationary environments.

The fuzzy or soft $k$-means objective is a popular generalization of the well-known $k$-means problem, extending the clustering capability of the $k$-means to datasets that are uncertain, vague and otherwise hard to cluster. In this paper, we propose a semi-supervised active clustering framework, where the learner is allowed to interact with an oracle (domain expert), asking for the similarity between a certain set of chosen items. We study the query and computational complexities of clustering in this framework. We prove that having a few of such similarity queries enables one to get a polynomial-time approximation algorithm to an otherwise conjecturally NP-hard problem. In particular, we provide algorithms for fuzzy clustering in this setting that ask $O(\mathsf{poly}(k)\log n)$ similarity queries and run with polynomial-time-complexity, where $n$ is the number of items. The fuzzy $k$-means objective is nonconvex, with $k$-means as a special case, and is equivalent to some other generic nonconvex problem such as non-negative matrix factorization. The ubiquitous Lloyd-type algorithms (or alternating-minimization algorithms) can get stuck at a local minima. Our results show that by making few similarity queries, the problem becomes easier to solve. Finally, we test our algorithms over real-world datasets, showing their effectiveness in real-world applications.

Recovery of support of a sparse vector from simple measurements is a widely studied problem, considered under the frameworks of compressed sensing, 1-bit compressed sensing, and more general single index models. We consider generalizations of this problem: mixtures of linear regressions, and mixtures of linear classifiers, where the goal is to recover supports of multiple sparse vectors using only a small number of possibly noisy linear, and 1-bit measurements respectively. The key challenge is that the measurements from different vectors are randomly mixed. Both of these problems have also received attention recently. In mixtures of linear classifiers, an observation corresponds to the side of the queried hyperplane a random unknown vector lies in; whereas in mixtures of linear regressions we observe the projection of a random unknown vector on the queried hyperplane. The primary step in recovering the unknown vectors from the mixture is to first identify the support of all the individual component vectors. In this work, we study the number of measurements sufficient for recovering the supports of all the component vectors in a mixture in both these models. We provide algorithms that use a number of measurements polynomial in $k, \log n$ and quasi-polynomial in $\ell$, to recover the support of all the $\ell$ unknown vectors in the mixture with high probability when each individual component is a $k$-sparse $n$-dimensional vector.

In the beautifully simple-to-state problem of trace reconstruction, the goal is to reconstruct an unknown binary string x given random “traces” of x where each trace is generated by deleting each coordinate of x independently with probability p1/4√{logn})) traces suffice for reconstructing arbitrary matrices. In the matrix version of the problem, each row and column of an unknown √n×√n matrix is deleted independently with probability p. Our results contrasts with the best known results for sequence reconstruction where the best known upper bound is exp(O(n1/3)). 2) An optimal result for random matrix reconstruction: we show that Θ(logn) traces are necessary and sufficient. This is in contrast to the problem for random sequences where there is a super-logarithmic lower bound and the best known upper bound is exp(O(log1/3n)). 3) We show that exp(O(k1/3log2/3n)) traces suffice to reconstruct k-sparse strings, providing an improvement over the best known sequence reconstruction results when k = o(n/log2n). 4) We show that poly(n) traces suffice if x is k-sparse and we additionally have a “separation” promise, specifically that the indices of 1's in x all differ by Ω(k logn).

Source coding is the canonical problem of data compression in information theory. In alocally encodablesource coding, each compressed bit depends on only few bits of the input. In this paper, we show that a recently popular model of semisupervised clustering is equivalent to locally encodable source coding. In this model, the task is to perform multiclass labeling of unlabeled elements. At the beginning, we can ask in parallel a set of simple queries to an oracle who provides (possibly erroneous) binary answers to the queries. The queries cannot involve more than two (or a fixed constant number of) elements. Now the labeling of all the elements (or clustering) must be performed based on the noisy query answers. The goal is to recover all the correct labelings while minimizing the number of such queries. The equivalence to locally encodable source codes leads us to find lower bounds on the number of queries required in variety of scenarios. We provide querying schemes based on pairwise ‘same cluster’ queries - and pairwise AND queries, and show provable performance guarantees for each of the schemes.

We present two different approaches for parameter learning in several mixture models in one dimension. Our first approach uses complex-analytic methods and applies to Gaussian mixtures with shared variance, binomial mixtures with shared success probability, and Poisson mixtures, among others. An example result is that $\exp(O(N^{1/3}))$ samples suffice to exactly learn a mixture of $k Cite this Paper BibTeX @InProceedings{pmlr-v117-krishnamurthy20a, title = {Algebraic and Analytic Approaches for Parameter Learning in Mixture Models}, author = {Krishnamurthy, Akshay and Mazumdar, Arya and McGregor, Andrew and Pal, Soumyabrata}, booktitle = {Proceedings of the 31st International Conference on Algorithmic Learning Theory}, pages = {468--489}, year = {2020}, editor = {Kontorovich, Aryeh and Neu, Gergely}, volume = {117}, series = {Proceedings of Machine Learning Research}, month = {08 Feb--11 Feb}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v117/krishnamurthy20a/krishnamurthy20a.pdf}, url = {https://proceedings.mlr.press/v117/krishnamurthy20a.html}, abstract = {We present two different approaches for parameter learning in several mixture models in one dimension. Our first approach uses complex-analytic methods and applies to Gaussian mixtures with shared variance, binomial mixtures with shared success probability, and Poisson mixtures, among others. An example result is that $\exp(O(N^{1/3}))$ samples suffice to exactly learn a mixture of $k Copy to Clipboard Download Endnote %0 Conference Paper %T Algebraic and Analytic Approaches for Parameter Learning in Mixture Models %A Akshay Krishnamurthy %A Arya Mazumdar %A Andrew McGregor %A Soumyabrata Pal %B Proceedings of the 31st International Conference on Algorithmic Learning Theory %C Proceedings of Machine Learning Research %D 2020 %E Aryeh Kontorovich %E Gergely Neu %F pmlr-v117-krishnamurthy20a %I PMLR %P 468--489 %U https://proceedings.mlr.press/v117/krishnamurthy20a.html %V 117 %X We present two different approaches for parameter learning in several mixture models in one dimension. Our first approach uses complex-analytic methods and applies to Gaussian mixtures with shared variance, binomial mixtures with shared success probability, and Poisson mixtures, among others. An example result is that $\exp(O(N^{1/3}))$ samples suffice to exactly learn a mixture of $k Copy to Clipboard Download APA Krishnamurthy, A., Mazumdar, A., McGregor, A. & Pal, S.. (2020). Algebraic and Analytic Approaches for Parameter Learning in Mixture Models. Proceedings of the 31st International Conference on Algorithmic Learning Theory, in Proceedings of Machine Learning Research 117:468-489 Available from https://proceedings.mlr.press/v117/krishnamurthy20a.html. Copy to Clipboard Download Related Material Download PDF This site last compiled Sun, 05 Jul 2026 14:49:43 +0000 Github Account Copyright © The authors and PMLR 2026. MLResearchPress

Mixture of linear regressions is a popular learning theoretic model that is used widely to represent heterogeneous data. In the simplest form, this model assumes that the labels are generated from either of two different linear models and mixed together. Recent works of Yin et al. and Krishnamurthy et al., 2019, focus on an experimental design setting of model recovery for this problem. It is assumed that the features can be designed and queried with to obtain their label. When queried, an oracle randomly selects one of the two different sparse linear models and generates a label accordingly. How many such oracle queries are needed to recover both of the models simultaneously? This question can also be thought of as a generalization of the well-known compressed sensing problem (Candès and Tao, 2005, Donoho, 2006). In this work we address this query complexity problem and provide efficient algorithms that improves on the previously best known results.

In the problem of learning a mixture of linear classifiers, the aim is to learn a collection of hyperplanes from a sequence of binary responses. Each response is a result of querying with a vector and indicates the side of a randomly chosen hyperplane from the collection the query vector belong to. This model is quite rich while dealing with heterogeneous data with categorical labels and has only been studied in some special settings. We look at a hitherto unstudied problem of query complexity upper bound of recovering all the hyperplanes, especially for the case when the hyperplanes are sparse. This setting is a natural generalization of the extreme quantization problem known as 1-bit compressed sensing. Suppose we have a set of l unknown k-sparse vectors. We can query the set with another vector a, to obtain the sign of the inner product of a and a randomly chosen vector from the l-set. How many queries are sufficient to identify all the l unknown vectors? This question is significantly more challenging than both the basic 1-bit compressed sensing problem (i.e., l = 1 case) and the analogous regression problem (where the value instead of the sign is provided). We provide rigorous query complexity results (with efficient algorithms) for this problem.

Recently Ermon et al. (2013) pioneered a way to practically compute approximations to large scale counting or discrete integration problems by using random hashes. The hashes are used to reduce the counting problem into many separate discrete optimization problems. The optimization problems then can be solved by an NP-oracle such as commercial SAT solvers or integer linear programming (ILP) solvers. In particular, Ermon et al. showed that if the domain of integration is $\{0,1\}^n$ then it is possible to obtain a solution within a factor of $16$ of the optimal (16-approximation) by this technique. In many crucial counting tasks, such as computation of partition function of ferromagnetic Potts model, the domain of integration is naturally $\{0,1,…, q-1\}^n, q>2$, the hypergrid. The straightforward extension of Ermon et al.’s method allows a $q^2$-approximation for this problem. For large values of $q$, this is undesirable. In this paper, we show an improved technique to obtain an approximation factor of $4+O(1/q^2)$ to this problem. We are able to achieve this by using an idea of optimization over multiple bins of the hash functions, that can be easily implemented by inequality constraints, or even in unconstrained way. The NP oracle in this setting can be simulated by using an ILP solver as in Ermon et. al. We provide simulation results to support the theoretical guarantees of our algorithms.

Random geometric graph (Gilbert, 1961) is a basic model of random graphs for spatial networks proposed shortly after the introduction of the Erdős-Rényi random graphs. The geometric block model (GBM) is a probabilistic model for community detection defined over random geometric graphs (RGG) similar in spirit to the popular stochastic block model which is defined over Erdős-Rényi random graphs. The GBM naturally inherits many desirable properties of RGGs such as transitivity ("friends having common friends') and has been shown to model many real-world networks better than the stochastic block model. Analyzing the properties of a GBM requires new tools and perspectives to handle correlation in edge formation. In this paper, we study the necessary and sufficient conditions for community recovery over GBM in the connectivity regime. We provide efficient algorithms that recover the communities exactly with high probability and match the lower bound within a small constant factor. This requires us to prove new connectivity results for vertex-random graphs or random annulus graphs which are natural generalizations of random geometric graphs. A vertex-random graph is a model of random graphs where the randomness lies in the vertices as opposed to an Erdős-Rényi random graph where the randomness lies in the edges. A vertex-random graph G(n, [r_1, r_2]), 0 <=r_1 0 vs when r_1=0 (RGG). We then extend the connectivity results to high dimensions. These results play a crucial role in analyzing the GBM.

Overlapping clusters are common in models of many practical data-segmentation applications. Suppose we are given $n$ elements to be clustered into $k$ possibly overlapping clusters, and an oracle that can interactively answer queries of the form ``do elements $u$ and $v$ belong to the same cluster?'' The goal is to recover the clusters with minimum number of such queries. This problem has been of recent interest for the case of disjoint clusters. In this paper, we look at the more practical scenario of overlapping clusters, and provide upper bounds (with algorithms) on the sufficient number of queries. We provide algorithmic results under both arbitrary (worst-case) and statistical modeling assumptions. Our algorithms are parameter free, efficient, and work in the presence of random noise. We also derive information-theoretic lower bounds on the number of queries needed, proving that our algorithms are order optimal. Finally, we test our algorithms over both synthetic and real-world data, showing their practicality and effectiveness.

In the problem of learning mixtures of linear regressions, the goal is to learn a col-lection of signal vectors from a sequence of (possibly noisy) linear measurements,where each measurement is evaluated on an unknown signal drawn uniformly fromthis collection. This setting is quite expressive and has been studied both in termsof practical applications and for the sake of establishing theoretical guarantees. Inthis paper, we consider the case where the signal vectors aresparse; this generalizesthe popular compressed sensing paradigm. We improve upon the state-of-the-artresults as follows: In the noisy case, we resolve an open question of Yin et al. (IEEETransactions on Information Theory, 2019) by showing how to handle collectionsof more than two vectors and present the first robust reconstruction algorithm, i.e.,if the signals are not perfectly sparse, we still learn a good sparse approximationof the signals. In the noiseless case, as well as in the noisy case, we show how tocircumvent the need for a restrictive assumption required in the previous work. Ourtechniques are quite different from those in the previous work: for the noiselesscase, we rely on a property of sparse polynomials and for the noisy case, we providenew connections to learning Gaussian mixtures and use ideas from the theory of

To capture the inherent geometric features of many community detection problems, we propose to use a new random graph model of communities that we call a Geometric Block Model. The geometric block model generalizes the random geometric graphs in the same way that the well-studied stochastic block model generalizes the Erdös-Renyi random graphs. It is also a natural extension of random community models inspired by the recent theoretical and practical advancement in community detection. While being a topic of fundamental theoretical interest, our main contribution is to show that many practical community structures are better explained by the geometric block model. We also show that a simple triangle-counting algorithm to detect communities in the geometric block model is near-optimal. Indeed, even in the regime where the average degree of the graph grows only logarithmically with the number of vertices (sparse-graph), we show that this algorithm performs extremely well, both theoretically and practically. In contrast, the triangle-counting algorithm is far from being optimum for the stochastic block model. We simulate our results on both real and synthetic datasets to show superior performance of both the new model as well as our algorithm.

Source coding is the canonical problem of data compression in information theory. In a locally encodable source coding, each compressed bit depends on only few bits of the input. In this paper, we show that a recently popular model of semisupervised clustering is equivalent to locally encodable source coding. In this model, the task is to perform multiclass labeling of unlabeled elements. At the beginning, we can ask in parallel a set of simple queries to an oracle who provides (possibly erroneous) binary answers to the queries. The queries cannot involve more than two (or a fixed constant number $\Delta$ of) elements. Now the labeling of all the elements (or clustering) must be performed based on the (noisy) query answers. The goal is to recover all the correct labelings while minimizing the number of such queries. The equivalence to locally encodable source codes leads us to find lower bounds on the number of queries required in variety of scenarios. We are also able to show fundamental limitations of pairwise `same cluster' queries - and propose pairwise AND queries, that provably performs better in many situations.