arXiv subjects

Aritra Konar

Publications and source records attributed to Aritra Konar.

14 recordsLinked to original sources

arXiv2026

Differentially Private and Scalable Estimation of the Network Principal Component

Computing the principal component (PC) of the adjacency matrix of an undirected graph has several applications ranging from identifying key vertices for influence maximization and controlling diffusion processes, to discovering densely interconnected vertex subsets. However, many networked datasets are sensitive, which necessitates private computation of the PC for use in the aforementioned applications. Differential privacy has emerged as the gold standard in privacy-preserving data analysis, but existing DP algorithms for private PC suffer from low accuracy due to large noise injection or high complexity. Motivated by the large gap between the local and global sensitivities of the PC on real-graphs, we consider instance-specific mechanisms for privately computing the PC under edge-DP. These mechanisms guarantee privacy for all datasets, but provide good utility on ``well-behaved'' datasets by injecting smaller amounts of noise. More specifically, we consider the Propose-Test-Release (PTR) framework. Although computationally expensive in general, we design a novel approach for implementing a PTR variant in the same time as computation of a non-private PC, while offering good utility. Our framework tests in a differentially-private manner whether a given graph is ``well-behaved'' or not, and then tests whether its private to release a noisy PC with small noise. As a consequence, this also leads to the first DP algorithm for the Densest-$k$-subgraph problem, a key graph mining primitive. We run our method on diverse real-world networks, with the largest having 3 million vertices, and compare its utility to a pre-existing baseline based on the private power method (PPM). Although PTR requires a slightly larger privacy budget, on average, it achieves a 180-fold improvement in runtime over PPM.

cs.DS↗

arXiv2026

FairRARI: A Plug and Play Framework for Fairness-Aware PageRank

PageRank (PR) is a fundamental algorithm in graph machine learning tasks. Owing to the increasing importance of algorithmic fairness, we consider the problem of computing PR vectors subject to various group-fairness criteria based on sensitive attributes of the vertices. At present, principled algorithms for this problem are lacking - some cannot guarantee that a target fairness level is achieved, while others do not feature optimality guarantees. In order to overcome these shortcomings, we put forth a unified in-processing convex optimization framework, termed FairRARI, for tackling different group-fairness criteria in a ``plug and play'' fashion. Leveraging a variational formulation of PR, the framework computes fair PR vectors by solving a strongly convex optimization problem with fairness constraints, thereby ensuring that a target fairness level is achieved. We further introduce three different fairness criteria which can be efficiently tackled using FairRARI to compute fair PR vectors with the same asymptotic time-complexity as the original PR algorithm. Extensive experiments on real-world datasets showcase that FairRARI outperforms existing methods in terms of utility, while achieving the desired fairness levels across multiple vertex groups; thereby highlighting its effectiveness.

cs.LG↗

arXiv2026

On Densest $k$-Subgraph Mining and Diagonal Loading: Optimization Landscape and Finite-Step Exact Convergence Analysis

The Densest $k$-Subgraph (D$k$S) is a fundamental combinatorial problem known for its theoretical hardness and breadth of applications. Recently, Lu et al. (AAAI 2025) introduced a penalty-based non-convex relaxation that achieves promising empirical performance; however, a rigorous theoretical understanding of its success remains unclear. In this work, we bridge this gap by providing a comprehensive theoretical analysis. We first establish the tightness of the relaxation, ensuring that the global maximum values of the original combinatorial problem and the relaxed problem coincide. Then we reveal the benign geometry of the optimization landscape by proving a strict dichotomy of stationary points: all integral stationary points are local maximizers, whereas all non-integral stationary points are strict saddles with explicit positive curvature. We propose a saddle-escaping Frank--Wolfe algorithm and prove that it achieves exact convergence to an integral local maximizer in a finite number of steps.

cs.SI↗

arXiv2025

A Scalable and Exact Relaxation for Densest $k$-Subgraph via Error Bounds

Given an undirected graph and a size parameter $k$, the Densest $k$-Subgraph (D$k$S) problem extracts the subgraph on $k$ vertices with the largest number of induced edges. While D$k$S is NP--hard and difficult to approximate, penalty-based continuous relaxations of the problem have recently enjoyed practical success for real-world instances of D$k$S. In this work, we propose a scalable and exact continuous penalization approach for D$k$S using the error bound principle, which enables the design of suitable penalty functions. Notably, we develop new theoretical guarantees ensuring that both the global and local optima of the penalized problem match those of the original problem. The proposed penalized reformulation enables the use of first-order continuous optimization methods. In particular, we develop a non-convex proximal gradient algorithm, where the non-convex proximal operator can be computed in closed form, resulting in low per-iteration complexity. We also provide convergence analysis of the algorithm. Experiments on large-scale instances of the D$k$S problem and one of its variants, the Densest ($k_1, k_2$) Bipartite Subgraph (D$k_1k_2$BS) problem, demonstrate that our method achieves a favorable balance between computation cost and solution quality.

eess.SP↗

arXiv2025

The Vertex-Attribute-Constrained Densest $k$-Subgraph Problem

Dense subgraph mining is a fundamental technique in graph mining, commonly applied in fraud detection, community detection, product recommendation, and document summarization. In such applications, we are often interested in identifying communities, recommendations, or summaries that reflect different constituencies, styles or genres, and points of view. For this task, we introduce a new variant of the Densest $k$-Subgraph (D$k$S) problem that incorporates the attribute values of vertices. The proposed Vertex-Attribute-Constrained Densest $k$-Subgraph (VAC-D$k$S) problem retains the NP-hardness and inapproximability properties of the classical D$k$S. Nevertheless, we prove that a suitable continuous relaxation of VAC-D$k$S is tight and can be efficiently tackled using a projection-free Frank--Wolfe algorithm. We also present an insightful analysis of the optimization landscape of the relaxed problem. Extensive experimental results demonstrate the effectiveness of our proposed formulation and algorithm, and its ability to scale up to large graphs. We further elucidate the properties of VAC-D$k$S versus classical D$k$S in a political network mining application, where VAC-D$k$S identifies a balanced and more meaningful set of politicians representing different ideological camps, in contrast to the classical D$k$S solution which is unbalanced and rather mundane.

cs.SI↗

arXiv2025

Fairness-Aware Dense Subgraph Discovery

Dense subgraph discovery (DSD) is a key graph mining primitive with myriad applications including finding densely connected communities which are diverse in their vertex composition. In such a context, it is desirable to extract a dense subgraph that provides fair representation of the diverse subgroups that constitute the vertex set while incurring a small loss in terms of subgraph density. Existing methods for promoting fairness in DSD have important limitations - the associated formulations are NP-hard in the worst case and they do not provide flexible notions of fairness, making it non-trivial to analyze the inherent trade-off between density and fairness. In this paper, we introduce two tractable formulations for fair DSD, each offering a different notion of fairness. Our methods provide a structured and flexible approach to incorporate fairness, accommodating varying fairness levels. We introduce the fairness-induced relative loss in subgraph density as a price of fairness measure to quantify the associated trade-off. We are the first to study such a notion in the context of detecting fair dense subgraphs. Extensive experiments on real-world datasets demonstrate that our methods not only match but frequently outperform existing solutions, sometimes incurring even less than half the subgraph density loss compared to prior art, while achieving the target fairness levels. Importantly, they excel in scenarios that previous methods fail to adequately handle, i.e., those with extreme subgroup imbalances, highlighting their effectiveness in extracting fair and dense solutions.

cs.SI↗

arXiv2023

Minimizing low-rank models of high-order tensors: Hardness, span, tight relaxation, and applications

We consider the problem of finding the smallest or largest entry of a tensor of order N that is specified via its rank decomposition. Stated in a different way, we are given N sets of R-dimensional vectors and we wish to select one vector from each set such that the sum of the Hadamard product of the selected vectors is minimized or maximized. We show that this fundamental tensor problem is NP-hard for any tensor rank higher than one, and polynomial-time solvable in the rank-one case. We also propose a continuous relaxation and prove that it is tight for any rank. For low-enough ranks, the proposed continuous reformulation is amenable to low-complexity gradient-based optimization, and we propose a suite of gradient-based optimization algorithms drawing from projected gradient descent, Frank-Wolfe, or explicit parametrization of the relaxed constraints. We also show that our core results remain valid no matter what kind of polyadic tensor model is used to represent the tensor of interest, including Tucker, HOSVD/MLSVD, tensor train, or tensor ring. Next, we consider the class of problems that can be posed as special instances of the problem of interest. We show that this class includes the partition problem (and thus all NP-complete problems via polynomial-time transformation), integer least squares, integer linear programming, integer quadratic programming, sign retrieval (a special kind of mixed integer programming / restricted version of phase retrieval), and maximum likelihood decoding of parity check codes. We demonstrate promising experimental results on a number of hard problems, including state-of-art performance in decoding low density parity check codes and general parity check codes.

eess.SP↗

arXiv2021

Exploring the Subgraph Density-Size Trade-off via the Lovász Extension

Given an undirected graph, the Densest-k-Subgraph problem (DkS) seeks to find a subset of k vertices such that the sum of the edge weights in the corresponding subgraph is maximized. The problem is known to be NP-hard, and is also very difficult to approximate, in the worst-case. In this paper, we present a new convex relaxation for the problem. Our key idea is to reformulate DkS as minimizing a submodular function subject to a cardinality constraint. Exploiting the fact that submodular functions possess a convex, continuous extension (known as the Lovász extension), we propose to minimize the Lovász extension over the convex hull of the cardinality constraints. Although the Lovász extension of a submodular function does not admit an analytical form in general, for DkS we show that it does. We leverage this result to develop a highly scalable algorithm based on the Alternating Direction Method of Multipliers (ADMM) for solving the relaxed problem. Coupled with a pair of fortuitously simple rounding schemes, we demonstrate that our approach outperforms existing baselines on real-world graphs and can yield high quality sub-optimal solutions which typically are a posteriori no worse than 65-80\% of the optimal density.

cs.SI↗

arXiv2020

PHASED: Phase-Aware Submodularity-Based Energy Disaggregation

Energy disaggregation is the task of discerning the energy consumption of individual appliances from aggregated measurements, which holds promise for understanding and reducing energy usage. In this paper, we propose PHASED, an optimization approach for energy disaggregation that has two key features: PHASED (i) exploits the structure of power distribution systems to make use of readily available measurements that are neglected by existing methods, and (ii) poses the problem as a minimization of a difference of submodular functions. We leverage this form by applying a discrete optimization variant of the majorization-minimization algorithm to iteratively minimize a sequence of global upper bounds of the cost function to obtain high-quality approximate solutions. PHASED improves the disaggregation accuracy of state-of-the-art models by up to 61% and achieves better prediction on heavy load appliances.

eess.SP↗

arXiv2020

Mining Large Quasi-cliques with Quality Guarantees from Vertex Neighborhoods

Mining dense subgraphs is an important primitive across a spectrum of graph-mining tasks. In this work, we formally establish that two recurring characteristics of real-world graphs, namely heavy-tailed degree distributions and large clustering coefficients, imply the existence of substantially large vertex neighborhoods with high edge-density. This observation suggests a very simple approach for extracting large quasi-cliques: simply scan the vertex neighborhoods, compute the clustering coefficient of each vertex, and output the best such subgraph. The implementation of such a method requires counting the triangles in a graph, which is a well-studied problem in graph mining. When empirically tested across a number of real-world graphs, this approach reveals a surprise: vertex neighborhoods include maximal cliques of non-trivial sizes, and the density of the best neighborhood often compares favorably to subgraphs produced by dedicated algorithms for maximizing subgraph density. For graphs with small clustering coefficients, we demonstrate that small vertex neighborhoods can be refined using a local-search method to ``grow'' larger cliques and near-cliques. Our results indicate that contrary to worst-case theoretical results, mining cliques and quasi-cliques of non-trivial sizes from real-world graphs is often not a difficult problem, and provides motivation for further work geared towards a better explanation of these empirical successes.

cs.SI↗

arXiv2020

Cell-Edge Detection via Selective Cooperation and Generalized Canonical Correlation

Improving the uplink quality of service for users located around the boundaries between cells is a key challenge in LTE systems. Relying on power control, existing approaches throttle the rates of cell-center users, while multi-user detection requires accurate channel estimates for the cell-edge users, which is another challenge due to their low received signal-to-noise ratio (SNR). Utilizing the fact that cell-edge user signals are weak but common (received at roughly equal power) at different base stations (BSs), this paper establishes a connection between cell-edge user detection and generalized canonical correlation analysis (GCCA). It puts forth a GCCA-based method that leverages selective BS cooperation to recover the cell-edge user signal subspace even at low SNR. The cell-edge user signals can then be extracted from the resulting mixture via algebraic signal processing techniques. The paper includes theoretical analysis showing why GCCA recovers the correct subspace containing the cell-edge user signals under mild conditions. The proposed method can also identify the number of cell-edge users in the system, i.e., the common subspace dimension. Simulations reveal significant performance improvement relative to various multiuser detection techniques. Cell-edge detection performance is further studied as a function of how many / which BSs are selected, and it is shown that using the closest three BS is always the best choice.

eess.SP↗

arXiv2020

Fast Algorithms for Joint Multicast Beamforming and Antenna Selection in Massive MIMO

Massive MIMO is currently a leading physical layer technology candidate that can dramatically enhance throughput in 5G systems, for both unicast and multicast transmission modalities. As antenna elements are becoming smaller and cheaper in the mmW range compared to radio frequency (RF) chains, it is crucial to perform antenna selection at the transmitter, such that the available RF chains are switched to an appropriate subset of antennas. This paper considers the joint problem of multicast beamforming and antenna selection for a single multicast group in massive MIMO systems. The prior state-of-art for this problem relies on semi-definite relaxation (SDR), which cannot scale up to the massive MIMO regime. A successive convex approximation (SCA) based approach is proposed to tackle max-min fair joint multicast beamforming and antenna selection. The key idea of SCA is to successively approximate the non-convex problem by a class of non-smooth, convex optimization problems. Two fast and memory efficient first-order methods are proposed to solve each SCA subproblem. Simulations demonstrate that the proposed algorithms outperform the existing state-of-art approach in terms of solution quality and run time, in both traditional and especially in massive MIMO settings.

eess.SP↗

arXiv2018

Mirror-Prox SCA Algorithm for Multicast Beamforming and Antenna Selection

This paper considers the (NP-)hard problem of joint multicast beamforming and antenna selection. Prior work has focused on using Semi-Definite relaxation (SDR) techniques in an attempt to obtain a high quality sub-optimal solution. However, SDR suffers from the drawback of having high computational complexity, as SDR lifts the problem to higher dimensional space, effectively squaring the number of variables. This paper proposes a high performance, low complexity Successive Convex Approximation (SCA) algorithm for max-min SNR "fair" joint multicast beamforming and antenna selection under a sum power constraint. The proposed approach relies on iteratively approximating the non-convex objective with a series of non-smooth convex subproblems, and then, a first order-based method called Saddle Point Mirror-Prox (SP-MP) is used to compute optimal solutions for each SCA subproblem. Simulations reveal that the SP-MP SCA algorithm provides a higher quality and lower complexity solution compared to the one obtained using SDR.

cs.IT↗

arXiv2014

Feasible Point Pursuit and Successive Approximation of Non-convex QCQPs

Quadratically constrained quadratic programs (QCQPs) have a wide range of applications in signal processing and wireless communications. Non-convex QCQPs are NP-hard in general. Existing approaches relax the non-convexity using semi-definite relaxation (SDR) or linearize the non-convex part and solve the resulting convex problem. However, these techniques are seldom successful in even obtaining a feasible solution when the QCQP matrices are indefinite. In this paper, a new feasible point pursuit successive convex approximation (FPP-SCA) algorithm is proposed for non-convex QCQPs. FPP-SCA linearizes the non-convex parts of the problem as conventional SCA does, but adds slack variables to sustain feasibility, and a penalty to ensure slacks are sparingly used. When FPP-SCA is successful in identifying a feasible point of the non-convex QCQP, convergence to a Karush-Kuhn-Tucker (KKT) point is thereafter ensured. Simulations show the effectiveness of our proposed algorithm in obtaining feasible and near-optimal solutions, significantly outperforming existing approaches.

math.OC↗