Tight Lower Bounds for Data-Dependent Locality-Sensitive Hashing

We prove a tight lower bound for the exponent $\rho$ for data-dependent Locality-Sensitive Hashing schemes, recently used to design efficient solutions for the $c$-approximate nearest neighbor search. In particular, our lower bound matches the bound of $\rho\le \frac{1}{2c-1}+o(1)$ for the $\ell_1$ space, obtained via the recent algorithm from [Andoni-Razenshteyn, STOC'15]. In recent years it emerged that data-dependent hashing is strictly superior to the classical Locality-Sensitive Hashing, when the hash function is data-independent. In the latter setting, the best exponent has been already known: for the $\ell_1$ space, the tight bound is $\rho=1/c$, with the upper bound from [Indyk-Motwani, STOC'98] and the matching lower bound from [O'Donnell-Wu-Zhou, ITCS'11]. We prove that, even if the hashing is data-dependent, it must hold that $\rho\ge \frac{1}{2c-1}-o(1)$. To prove the result, we need to formalize the exact notion of data-dependent hashing that also captures the complexity of the hash functions (in addition to their collision properties). Without restricting such complexity, we would allow for obviously infeasible solutions such as the Voronoi diagram of a dataset. To preclude such solutions, we require our hash functions to be succinct. This condition is satisfied by all the known algorithmic results.Comment: 16 pages, no figure

Restricted Isometry Property for General p-Norms

The Restricted Isometry Property (RIP) is a fundamental property of a matrix which enables sparse recovery. Informally, an $m \times n$ matrix satisfies RIP of order $k$ for the $\ell_p$ norm, if $\|Ax\|_p \approx \|x\|_p$ for every vector $x$ with at most $k$ non-zero coordinates. For every $1 \leq p < \infty$ we obtain almost tight bounds on the minimum number of rows $m$ necessary for the RIP property to hold. Prior to this work, only the cases $p = 1$, $1 + 1 / \log k$, and $2$ were studied. Interestingly, our results show that the case $p = 2$ is a "singularity" point: the optimal number of rows $m$ is $\widetilde{\Theta}(k^{p})$ for all $p\in [1,\infty)\setminus \{2\}$, as opposed to $\widetilde{\Theta}(k)$ for $k=2$. We also obtain almost tight bounds for the column sparsity of RIP matrices and discuss implications of our results for the Stable Sparse Recovery problem.Comment: An extended abstract of this paper is to appear at the 31st International Symposium on Computational Geometry (SoCG 2015

Lower Bounds on Time-Space Trade-Offs for Approximate Near Neighbors

We show tight lower bounds for the entire trade-off between space and query time for the Approximate Near Neighbor search problem. Our lower bounds hold in a restricted model of computation, which captures all hashing-based approaches. In articular, our lower bound matches the upper bound recently shown in [Laarhoven 2015] for the random instance on a Euclidean sphere (which we show in fact extends to the entire space $\mathbb{R}^d$ using the techniques from [Andoni, Razenshteyn 2015]). We also show tight, unconditional cell-probe lower bounds for one and two probes, improving upon the best known bounds from [Panigrahy, Talwar, Wieder 2010]. In particular, this is the first space lower bound (for any static data structure) for two probes which is not polynomially smaller than for one probe. To show the result for two probes, we establish and exploit a connection to locally-decodable codes.Comment: 47 pages, 2 figures; v2: substantially revised introduction, lots of small corrections; subsumed by arXiv:1608.03580 [cs.DS] (along with arXiv:1511.07527 [cs.DS]

Centre for Discrete Mathematics and Theoretical Computer ScienceNot Every Domain of a Plain Decompressor Contains the Domain of a Prefix-Free One

question about the relation between plain and prefix Kolmogorov complexities (see their paper in DLT 2008 conference proceedings): does the domain of every optimal decompressor contain the domain of some optimal prefix-free decompressor? In this paper we provide a negative answer to this question.