A novel reinterpretation of the low-degree moments of the truncated Gaussian in terms of a relative truncation parameter that uniquely determines the parameters of the untruncated Gaussian and enables direct parameter recovery.
Abstract
We study the fundamental problem of learning a high-dimensional Gaussian truncated to an unknown halfspace. Lee, Mehrotra and Zampetakis (FOCS'24) recently obtained the first polynomial time algorithm for this problem, but their resulting sample and time complexity bounds are not optimal. Under non-trivial truncation, for any target accuracy $\varepsilon>0$ and dimension $d$ we give an efficient algorithm that uses $n = \tilde{O}(d^2/\varepsilon^2)$ samples and learns the underlying Gaussian to error $\varepsilon$ in total variation distance. Our algorithm is also fast: its runtime is dominated by the cost of computing the empirical covariance matrix. Both our sample and time complexity are optimal in terms of $d$ and $\varepsilon$ even without truncation: in this regard, we can learn a Gaussian under halfspace truncation for free. The key ingredient behind our result is a novel reinterpretation of the low-degree moments of the truncated Gaussian in terms of a relative truncation parameter. This relative truncation parameter uniquely determines the parameters of the untruncated Gaussian and enables direct parameter recovery. This reinterpretation allows us to circumvent the time intensive projected stochastic gradient descent procedure that is widely used in learning under truncation.
We describe a simple rejection-sampling-based algorithm to perform length-squared sampling on an $n \times n$ positive-semidefinite (psd) matrix: that is, to sample a column with probability proportional to its squared $\ell_2$-norm. The algorithm runs in just $O(n)$ expected time, which is significantly sublinear in the input matrix size. The runtime is optimal, even when the input is assumed to be diagonal. Our result has several applications. Length-squared sampling is used by a number of sublinear time algorithms for matrix problems, like low-rank approximation and eigenvalue approximation. Often, it is assumed that the algorithm is given access to the matrix column norms, and thus can perform length-squared sampling efficiently. Our result shows that, at least for psd matrices, we can remove this assumption. We also discuss an application to an asymptotically optimal algorithm for estimating the Frobenius norm of a psd matrix to relative error. Finally, we show that our sampling algorithm yields a very simple sublinear time algorithm for the robust psd low-rank approximation problem introduced by Bakshi et al. (FOCS, 2020), which nearly matches the more complex method developed there.
Rajarshi Bhattacharjee, Ethan N. Epperly, Cameron Musco et al.· 0 citations
We consider a mixture of at most $k$ unit-covariance Gaussians in $\mathbb{R}^d$ whose means belong to a fixed-radius ball, with no separation or minimum-weight condition. Doss, Wu, Yang and Zhou (2023) proved that the minimax Hellinger risk is of order $\sqrt{d/n}\wedge 1$ and constructed a proper polynomial-time estimator with the slower general bound $(d/n)^{1/4}$; obtaining the sharp rate in polynomial time for fixed $k\geq 3$ was left open. We resolve this question. The key device is a moment-fiber range finder. A second-moment subspace controls the energy missed by projection. We then estimate finitely many one-free-index Hermite contractions. These vector-valued contractions recover every tensor component containing exactly one missed direction at the sharp $\sqrt{d/n}$ scale. Every remaining term contains at least two missed factors and is therefore controlled by the residual second-moment energy. The resulting subspace has dimension depending only on $k$. Exhaustive moment fitting in this constant-dimensional space produces a proper mixture and, together with the dimension-free moment characterization of Gaussian mixtures, achieves the optimal Hellinger rate in polynomial arithmetic time for every fixed $k$.
Principal component analysis (PCA) is optimal for the linear reconstruction of Gaussian data, a foundational property underlying its central role in algorithms and signal processing. Its nonlinear analogue, however, is notoriously subtle: in 2011, Mallat and Zeitouni conjectured that the Karhunen--Lo\`eve (KL) basis remains optimal even when the retained coordinates are chosen adaptively per sample, a property that would theoretically justify the ubiquitous pipeline of PCA followed by sparse thresholding. In this paper, we establish a $1+O(1/\sqrt{d})$-approximate version of the retained-energy form of the Mallat--Zeitouni conjecture, showing that the KL basis is within this factor of the optimal basis. This dimension-free comparison depends only on the number of retained coordinates and shows that the possible advantage of optimizing over all orthonormal bases vanishes as $d$ grows. It complements the universal-constant reconstruction-error comparison of Litvak and Tikhomirov (Ann. Appl. Probab., 2018), while providing a comparison naturally suited for algorithmic analysis. Our proof rests on a clean, conceptual reduction: we relax arbitrary rotations to a deterministic threshold bound via Schur--Horn majorization, and identify the remaining loss with the correlation gap of the rank-$d$ uniform matroid over Gaussian level sets.
Minbo Gao, Zhengfeng Ji, Chenghua Liu· 0 citations
Uniform stability is a classical tool for controlling the generalization error of a learning algorithm. Bousquet, Klochkov, and Zhivotovskiy (2020) showed that the problem can be reduced to a moment inequality for a sum of weakly interacting functions of independent random variables. Their bound contains an additional factor $\log n$, and they asked whether this factor can be removed. We answer this upper-bound question affirmatively. More specifically, let $Z=(Z_1,\ldots,Z_n)$ have independent coordinates and let $g_i(Z)$ satisfy $$ \mathbb E[g_i(Z)\mid Z_{-i}]=0, \qquad \left| \mathbb E[g_i(Z)\mid Z_i]\right|\le M, \qquad \forall i = \overline{1, n} $$ while changing any coordinate $Z_j$, $j\neq i$, changes $g_i$ by at most $\beta$ and $Z_{-i}$ denotes all coordinates except $Z_i$. We prove that, for every $p\ge2$, $$ \left\| \sum_{i=1}^n g_i(Z)\right\|_p \le 16pn\beta+M\sqrt{2pn}. $$ This removes the $\log n$ factor from the previous bound and matches the lower bound of Bousquet, Klochkov, and Zhivotovskiy up to universal constants in the range covered by their construction. Our proof first establishes the required estimate on the Rademacher cube, then transfers it to arbitrary product distributions by a two-copy randomization argument.
Abstract.
We propose an adaptive randomized truncation estimator for Krylov subspace methods that optimizes the trade-off between the solution variance and the computational cost while remaining unbiased. The estimator solves a constrained optimization problem to compute the truncation probabilities on the fly with minimal computational overhead. The problem has a closed-form solution when the improvement of the deterministic algorithm satisfies a diminishing returns property. We prove that obtaining the optimal adaptive truncation distribution is impossible in the general case. Without the diminishing return condition, our estimator provides a suboptimal but still unbiased solution. We present experimental results in Gaussian process (GP) hyperparameter training and competitive physics-informed neural networks problems to demonstrate the effectiveness of our approach.
Reproducibility of computational results. This paper has been awarded the “SIAM Reproducibility Badge: Code and data available” as a recognition that the authors have followed reproducibility principles valued by SISC and the scientific computing community. Code and data that allow readers to reproduce the results in this paper are available at https://github.com/RockyL7/AdaptivelySubsampledKrylov.jl and in the supplementary materials ( AdaptivelySubsampledKrylov_jl-master.zip [14.5KB]). [Formula: see text]
Binary Iterative Hard Thresholding (BIHT) is a simple, yet effective, greedy method for recovering a sparse vector from one-bit sign measurements. In its original form, BIHT performs a ``gradient-descent''step, followed by hard thresholding. A convergence analysis of this algorithm was left open in the introductory work of [Jac+11] and has remained unresolved for over a decade, with subsequent sharp analyses studying a normalized variant instead, that additionally projects every iterate onto the unit sphere. This paper resolves that gap and characterizes when per-iteration normalization is algorithmically necessary. In the noiseless setting, we prove a universal, sample-optimal convergence theorem for the original BIHT algorithm. Specifically, with $\widetilde O(s/\epsilon)$ measurements, a deterministic finite-time iterate has directional error at most $\epsilon$, simultaneously for every $s$-sparse unit vector. This matches the optimal sample dependence achieved by normalized BIHT in prior work. Thus, in the noiseless regime, per-iterate normalization is unnecessary for optimal recovery. Under sign corruptions, we prove a sharp separation. If at most a $\tau$ fraction of signs are flipped adversarially, then BIHT, without per-iterate normalization, still reaches the robust error floor at an early iterate with a matching $\widetilde O(s/\epsilon)$ sample complexity rate as its normalized variant. This recovery, however, is not stable. We prove a scalar lower bound showing that any nontrivial corruption pattern, even one that involves only one flipped sign together with one clean sign, forces the iterates to oscillate indefinitely. Consequently, no general last-iterate convergence theorem can hold for BIHT under sign corruptions, while its normalized surrogate provably escapes this instance.