arXiv subjects

Tomer Zilca

Publications and source records attributed to Tomer Zilca.

2 recordsLinked to original sources

arXiv2026

Approximating Uniform Random Rotations by Two-Block Structured Hadamard Rotations in High Dimensions

Uniform random rotations are a useful primitive in applications such as fast Johnson-Lindenstrauss embeddings, kernel approximation, communication-efficient learning, and recent AI compression pipelines, but they are computationally expensive to generate and apply in high dimensions. A common practical replacement is repeated structured random rotations built from Walsh-Hadamard transforms and random sign diagonals. Applying the structured random rotation twice has been shown empirically to be useful, but the supporting theory is still limited. In this paper we study the approximation quality achieved when using this two-block structured Hadamard rotation. Our results are both positive and negative. On the positive side, we prove that every fixed coordinate of the two-block transform converges uniformly, over all inputs, to the corresponding coordinate of a uniformly rotated vector, with an explicit Kolmogorov-distance bound of order $d^{-1/5}$. On the negative side, we prove an explicit lower bound on the Wasserstein distance between the full vector distributions, showing that the two-block transform is not a globally accurate surrogate for a uniform random rotation in the worst case. For the extremal input used in the lower bound, we also prove a matching asymptotic upper bound, showing that the lower-bound scale is sharp for that input. Taken together, the results identify a clear separation between one-dimensional marginal behavior, where approximation improves with dimension, and full high-dimensional geometry, where a nonvanishing discrepancy remains. This provides a partial theoretical explanation for the empirical success of structured Hadamard rotations in some algorithms, while also clarifying the limitations of treating them as drop-in replacements for true uniform random rotations.

cs.LG↗

arXiv2023

The secretary problem with items arriving according to a random permutation avoiding a pattern of length three

In the classical secretary problem, $n$ ranked items arrive one by one, and each item's rank relative to its predecessors is noted. The observer must select or reject each item as it arrives, with the object of selecting the item of highest rank. For $M_n\in\{0,1,\cdots, n-1\}$, let $\mathcal{S}(n,M_n)$ denote the strategy whereby the observer rejects the first $M_n$ items, and then selects the first later-arriving item whose rank is higher than that of any of the first $M_n$ items (if such an item exists). If the ranked items arrive in a uniformly random order, it is well-known that the limiting optimal probability of success is $\frac1e$, which occurs if $M_n\sim\frac ne$. It has been shown that when the ranked items arrive according to certain non-uniform distributions on the set of permutations, $\frac1e$ serves as a lower bound for the optimal probability. There is a fundamental reason for this phenomenon. We consider certain distributions for which that reason does not apply. We begin by noting a cooked-up class of distributions for which $\mathcal{S}(n,M)$ yields the lowest possible probability of success -- namely $\frac1n$, for all $M$. We then consider the uniform distribution over all permutations avoiding a particular pattern of length three. In the case of the pattern 231 or 132, for any choice of $M_n$, the strategy $\mathcal{S}(n,M_n)$ yields the very same probability of success; namely $\frac{n+1}{2(2n-1)}$, which gives a limiting probability of $\frac14$. For the pattern 213, the optimal strategy is obtained for $M\in\{0,1\}$, also yielding a limiting probability of $\frac14$. For the pattern 123, the optimal strategy is obtained for $M=1$, yielding a limiting probability of $\frac34$. For the other two patterns, 312 and 321, an optimal strategy will yield a limiting probability of at least $\frac7{16}$.

math.PR↗