arXiv subjects

Prashanti Anderson

Publications and source records attributed to Prashanti Anderson.

7 recordsLinked to original sources

arXiv2026

Entrywise Low-Rank Approximation and Matrix $p \rightarrow q$ Norms via Global Correlation Rounding

Given a matrix $A$, the goal of the entrywise low-rank approximation problem is to find $\operatorname{argmin} \|A-B\|_p$ over all rank-$k$ matrices $B$, where $\| \cdot \|_p$ is the entrywise $\ell_p$ norm. When $p = 2$ this well-studied problem is solved by the singular value decomposition, but for $p \neq 2$ the problem becomes computationally challenging. For every even $p > 2$ and every fixed $k$, we give the first polynomial-time approximation scheme for this problem, improving on the $(3 + \varepsilon)$ approximation of Ban, Bhattiprolu, Bringmann, Kolev, Lee, and Woodruff, the bi-criteria approximation of Woodruff and Yasuda, and the additive approximation scheme of Anderson, Bakshi, and Hopkins. Prior algorithmic approaches based on sketching and column selection, which yielded a polynomial-time approximation scheme in the $p < 2$ setting, face concrete barriers when $p > 2$. Instead, we use the Sherali-Adams hierarchy of convex programs, and in so doing suggest a methodology for how to use convex hierarchies to design polynomial-time approximation schemes for continuous optimization problems. We use the same algorithmic strategy to give a new family of additive approximation algorithms for matrix $p \rightarrow q$ norms, which are intimately related to small-set expansion and quantum information. In particular, we give the first nontrivial additive approximation algorithms in the regime $p < 2 < q$.

cs.DS↗

arXiv2026

Certifiable Near-Optimality: A Simple Framework for Unifying Search and Refutation for (Semi)random CSPs

A classical problem in average-case complexity is the study of random constraint satisfaction problems (CSPs). Random CSPs are traditionally studied in two different settings: refutation, where the instances are uniformly random and thus unsatisfiable with high probability, and search, where the instances are drawn from a planted model so that they are satisfiable. While there is no formal relationship between the refutation and search variants of random CSPs, known algorithms are strikingly similar with near-identical computational thresholds. In this work, we establish a formal relationship between the known algorithms for refutation and search by showing that in either case they achieve a stronger guarantee: they output an assignment $x$ along with a certificate $π$ that the fraction of constraints satisfied by $x$ is within some small $\varepsilon$ of the optimal assignment. We call this guarantee certifiable $\varepsilon$-optimality. As an application, we design new algorithms for a model of semirandom CSPs where the instance hypergraph (or scopes) is random, but the literal negation patterns are adversarially chosen and may depend on the hypergraph. For such CSPs, we give a family of algorithms that output certifiably $\varepsilon$-optimal solutions. We additionally study such semirandom CSPs in the "strong contamination model", where an adversary is allowed to corrupt an $O(δ)$-fraction of constraints after seeing the initial CSP. For such CSPs, we give an algorithm to output a certifiably $O(δ)$-optimal solution.

cs.DS↗

arXiv2026

A Simple Algorithm for Best Separable State

We study the best separable state problem (BSS), which asks for the maximum acceptance probability of a quantum measurement over unentangled states. In classical terms, the goal is to maximize $\langle(x \otimes y), M (x \otimes y)\rangle$ over unit vectors $x,y$ where $0 \preceq M \preceq I$; we call this value $\mathrm{BSS}(M)$. We study $\mathrm{BSS}$ in the "perfect completeness" regime, where given $M$ such that $\mathrm{BSS}(M) = 1$ the goal is to find the best possible solution $x,y$ -- this generalizes the problem of finding a rank-one matrix as close as possible to a given subspace of $\mathbb{R}^{n \times n}$ guaranteed to contain a rank-one matrix. The strongest known algorithmic guarantees for this problem are: (1) an algorithm which finds a solution with value $1-\varepsilon$ in time $\exp(\sqrt{n} (\log n)^{O(1)} / \varepsilon^2)$, due to Barak, Kothari, and Steurer, and (2) an algorithm which finds a solution with value $q/n$ in time roughly $n^{O(q)}$, due to Bhattiprolu, Ghosh, Guruswami, Lee, and Tulsiani. We give a much simpler approach to rounding the SoS relaxation, generalizing the canonical "global correlation rounding" technique, and obtain a better running time. Given $M$ with $\mathrm{BSS}(M) = 1$, our algorithm finds a solution with value $1-ε$ in time $n^{O(\sqrt{n/\varepsilon})}$, and a solution of value $q/n$ in time $n^{O(\sqrt q)}$. Using the same techniques, we prove a new variant of the "pinning lemma", a measure-decomposition theorem widely used in LP/SDP rounding, high-dimensional probability, and statistical physics, which we believe is of independent interest.

cs.DS↗

arXiv2026

Dimension Reduction via Sum-of-Squares and Improved Clustering Algorithms for Non-Spherical Mixtures

We develop a new approach for clustering non-spherical (i.e., arbitrary component covariances) Gaussian mixture models via a subroutine, based on the sum-of-squares method, that finds a low-dimensional separation-preserving projection of the input data. Our method gives a non-spherical analog of the classical dimension reduction, based on singular value decomposition, that, among several other applications, forms a key component of the celebrated spherical clustering algorithm of Vempala and Wang [VW04]. As applications, we obtain an algorithm to (1) cluster an arbitrary total-variation separated mixture of $k$ centered (i.e., zero-mean) Gaussians with $n\geq \operatorname{poly}(d) f(w_{\min}^{-1})$ samples and $\operatorname{poly}(n)$ time, and (2) cluster an arbitrary total-variation separated mixture of $k$ Gaussians with identical but arbitrary unknown covariance with $n \geq d^{O(\log w_{\min}^{-1})} f(w_{\min}^{-1})$ samples and $n^{O(\log w_{\min}^{-1})}$ time. Here, $w_{\min}$ is the minimum mixing weight of the input mixture, and $f$ does not depend on the dimension $d$. Our algorithms naturally extend to tolerating a dimension-independent fraction of arbitrary outliers. Before this work, the techniques in the state-of-the-art non-spherical clustering algorithms needed $d^{O(k)} f(w_{\min}^{-1})$ samples and time for clustering such mixtures. Our results may come as a surprise in the context of the $d^{Ω(k)}$ statistical query and sum-of-squares lower bounds [DKS17, DKPP24] for clustering non-spherical Gaussian mixtures. While these results are usually thought to rule out $d^{o(k)}$ cost algorithms for the problem, our results show that the lower bounds can in fact be circumvented for a remarkably general class of Gaussian mixtures.

cs.DS↗

arXiv2025

Faster MAX-CUT on Bounded Threshold Rank Graphs

We design new algorithms for approximating 2CSPs on graphs with bounded threshold rank, that is, whose normalized adjacency matrix has few eigenvalues larger than $\varepsilon$, smaller than $-\varepsilon$, or both. Unlike on worst-case graphs, 2CSPs on bounded threshold rank graphs can be $(1+O(\varepsilon))$-approximated efficiently. Prior approximation algorithms for this problem run in time exponential in the threshold rank and $1/\varepsilon$. Our algorithm has running time which is polynomial in $1/\varepsilon$ and exponential in the threshold rank of the label-extended graph, and near-linear in the input size. As a consequence, we obtain the first $(1+O(\varepsilon))$ approximation for MAX-CUT on bounded threshold rank graphs running in $\mathrm{poly}(1/\varepsilon)$ time. We also improve the state-of-the-art running time for 2CSPs on bounded threshold-rank graphs from polynomial in input size to near-linear via a new comparison inequality between the threshold rank of the label-extended graph and base graph. Our algorithm is a simple yet novel combination of subspace enumeration and semidefinite programming.

cs.DS↗

arXiv2025

Additive Approximation Schemes for Low-Dimensional Embeddings

We consider the task of fitting low-dimensional embeddings to high-dimensional data. In particular, we study the $k$-Euclidean Metric Violation problem ($\textsf{$k$-EMV}$), where the input is $D \in \mathbb{R}^{\binom{n}{2}}_{\geq 0}$ and the goal is to find the closest vector $X \in \mathbb{M}_{k}$, where $\mathbb{M}_k \subset \mathbb{R}^{\binom{n}{2}}_{\geq 0}$ is the set of all $k$-dimensional Euclidean metrics on $n$ points, and closeness is formulated as the following optimization problem, where $\| \cdot \|$ is the entry-wise $\ell_2$ norm: \[ \textsf{OPT}_{\textrm{EMV}} = \min_{X \in \mathbb{M}_{k} } \Vert D - X \Vert_2^2\,.\] Cayton and Dasgupta [CD'06] showed that this problem is NP-Hard, even when $k=1$. Dhamdhere [Dha'04] obtained a $O(\log(n))$-approximation for $\textsf{$1$-EMV}$ and leaves finding a PTAS for it as an open question (reiterated recently by Lee [Lee'25]). Although $\textsf{$k$-EMV}$ has been studied in the statistics community for over 70 years, under the name "multi-dimensional scaling", there are no known efficient approximation algorithms for $k > 1$, to the best of our knowledge. We provide the first polynomial-time additive approximation scheme for $\textsf{$k$-EMV}$. In particular, we obtain an embedding with objective value $\textsf{OPT}_{\textrm{EMV}} + \varepsilon \Vert D\Vert_2^2$ in $(n\cdot B)^{\mathsf{poly}(k, \varepsilon^{-1})}$ time, where each entry in $D$ can be represented by $B$ bits. We believe our algorithm is a crucial first step towards obtaining a PTAS for $\textsf{$k$-EMV}$. Our key technical contribution is a new analysis of correlation rounding for Sherali-Adams / Sum-of-Squares relaxations, tailored to low-dimensional embeddings. We also show that our techniques allow us to obtain additive approximation schemes for two related problems: a weighted variant of $\textsf{$k$-EMV}$ and $\ell_p$ low-rank approximation for $p>2$.

cs.DS↗

arXiv2025

Sample-Optimal Private Regression in Polynomial Time

We consider the task of privately obtaining prediction error guarantees in ordinary least-squares regression problems with Gaussian covariates (with unknown covariance structure). We provide the first sample-optimal polynomial time algorithm for this task under both pure and approximate differential privacy. We show that any improvement to the sample complexity of our algorithm would violate either statistical-query or information-theoretic lower bounds. Additionally, our algorithm is robust to a small fraction of arbitrary outliers and achieves optimal error rates as a function of the fraction of outliers. In contrast, all prior efficient algorithms either incurred sample complexities with sub-optimal dimension dependence, scaling with the condition number of the covariates, or obtained a polynomially worse dependence on the privacy parameters. Our technical contributions are two-fold: first, we leverage resilience guarantees of Gaussians within the sum-of-squares framework. As a consequence, we obtain efficient sum-of-squares algorithms for regression with optimal robustness rates and sample complexity. Second, we generalize the recent robustness-to-privacy framework [HKMN23, (arXiv:2212.05015)] to account for the geometry induced by the covariance of the input samples. This framework crucially relies on the robust estimators to be sum-of-squares algorithms, and combining the two steps yields a sample-optimal private regression algorithm. We believe our techniques are of independent interest, and we demonstrate this by obtaining an efficient algorithm for covariance-aware mean estimation, with an optimal dependence on the privacy parameters.

cs.DS↗