arXiv subjects

Prashanti Anderson

Publications and source records attributed to Prashanti Anderson.

6 recordsLinked to original sources

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 $1-\epsilon$ 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

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 establish a blueprint 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↗

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↗

arXiv2024

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 $d^{\Omega(k)}$ 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↗