Ria Stevens

Ria Stevens, Fangshuo Liao, Barbara Su et al.

We approach the Max-3-Cut problem through the lens of maximizing complex-valued quadratic forms and demonstrate that low-rank structure in the objective matrix can be exploited, leading to alternative algorithms to classical semidefinite programming (SDP) relaxations and heuristic techniques. We propose an algorithm for maximizing these quadratic forms over a domain of size $K$ that enumerates and evaluates a set of $O\left(n^{2r-1}\right)$ candidate solutions, where $n$ is the dimension of the matrix and $r$ represents the rank of an approximation of the objective. We prove that this candidate set is guaranteed to include the exact maximizer when $K=3$ (corresponding to Max-3-Cut) and the objective is low-rank, and provide approximation guarantees when the objective is a perturbation of a low-rank matrix. This construction results in a family of novel, inherently parallelizable and theoretically-motivated algorithms for Max-3-Cut. Extensive experimental results demonstrate that our approach achieves performance comparable to existing algorithms across a wide range of graphs, while being highly scalable.

Chen Dun, Qiutai Pan, Shikai Jin et al.

Determining the structure of a protein has been a decades-long open question. A protein's three-dimensional structure often poses nontrivial computation costs, when classical simulation algorithms are utilized. Advances in the transformer neural network architecture -- such as AlphaFold2 -- achieve significant improvements for this problem, by learning from a large dataset of sequence information and corresponding protein structures. Yet, such methods only focus on sequence information; other available prior knowledge, such as protein crystallography and partial structure of amino acids, could be potentially utilized. To the best of our knowledge, we propose the first transformer-based model that directly utilizes protein crystallography and partial structure information to predict the electron density maps of proteins. Via two new datasets of peptide fragments (2-residue and 15-residue) , we demonstrate our method, dubbed \texttt{CrysFormer}, can achieve accurate predictions, based on a much smaller dataset size and with reduced computation costs.

Maryam Aliakbarpour, Alireza Azizi, Ria Stevens

Independence testing is a fundamental problem in statistical inference: given samples from a joint distribution $p$ over multiple random variables, the goal is to determine whether $p$ is a product distribution or is $ε$-far from all product distributions in total variation distance. In the non-parametric finite-sample regime, this task is notoriously expensive, as the minimax sample complexity scales polynomially with the support size. In this work, we move beyond these worst-case limitations by leveraging the framework of \textit{augmented distribution testing}. We design independence testers that incorporate auxiliary, but potentially untrustworthy, predictive information. Our framework ensures that the tester remains robust, maintaining worst-case validity regardless of the prediction's quality, while significantly improving sample efficiency when the prediction is accurate. Our main contributions include: (i) a bivariate independence tester for discrete distributions that adaptively reduces sample complexity based on the prediction error; (ii) a generalization to the high-dimensional multivariate setting for testing the independence of $d$ random variables; and (iii) matching minimax lower bounds demonstrating that our testers achieve optimal sample complexity.

Maryam Aliakbarpour, Alireza Fallah, Swaha Roy et al.

We study statistical estimation under local differential privacy (LDP) when users may hold heterogeneous privacy levels and accuracy must be guaranteed with high probability. Departing from the common in-expectation analyses, and for one-dimensional and multi-dimensional mean estimation problems, we develop finite sample upper bounds in $\ell_2$-norm that hold with probability at least $1-β$. We complement these results with matching minimax lower bounds, establishing the optimality (up to constants) of our guarantees in the heterogeneous LDP regime. We further study distribution learning in $\ell_\infty$-distance, designing an algorithm with high-probability guarantees under heterogeneous privacy demands. Our techniques offer principled guidance for designing mechanisms in settings with user-specific privacy levels.

Maryam Aliakbarpour, Zhan Shi, Ria Stevens et al.

Estimating the density of a distribution from its samples is a fundamental problem in statistics. Hypothesis selection addresses the setting where, in addition to a sample set, we are given $n$ candidate distributions -- referred to as hypotheses -- and the goal is to determine which one best describes the underlying data distribution. This problem is known to be solvable very efficiently, requiring roughly $O(\log n)$ samples and running in $\tilde{O}(n)$ time. The quality of the output is measured via the total variation distance to the unknown distribution, and the approximation factor of the algorithm determines how large this distance is compared to the optimal distance achieved by the best candidate hypothesis. It is known that $α= 3$ is the optimal approximation factor for this problem. We study hypothesis selection under the constraint of differential privacy. We propose a differentially private algorithm in the central model that runs in nearly-linear time with respect to the number of hypotheses, achieves the optimal approximation factor, and incurs only a modest increase in sample complexity, which remains polylogarithmic in $n$. This resolves an open question posed by [Bun, Kamath, Steinke, Wu, NeurIPS 2019]. Prior to our work, existing upper bounds required quadratic time.