Embedding tables are used by machine learning systems to work with categorical features.In modern Recommendation Systems, these tables can be very large, necessitating the development of new methods for fitting them in memory, even during training.We suggest Clustered Compositional Embeddings (CCE) which combines clustering-based compression like quantization to codebooks with dynamic methods like The Hashing Trick and Compositional Embeddings [Shi et al., 2020].Experimentally CCE achieves the best of both worlds: The high compression rate of codebook-based quantization, but \emph{dynamically} like hashing-based methods, so it can be used during training.Theoretically, we prove that CCE is guaranteed to converge to the optimal codebook and give a tight bound for the number of iterations required.

A polyomino is a polygonal region with axis-parallel edges and corners of integral coordinates, which may have holes. In this paper, we consider planar tiling and packing problems with polyomino pieces and a polyomino container P . We give polynomial-time algorithms for deciding if P can be tiled with k × k squares for any fixed k which can be part of the input (that is, deciding if P is the union of a set of non-overlapping k × k squares) and for packing P with a maximum number of non-overlapping and axis-parallel 2 × 1 dominos, allowing rotations by 90°. As packing is more general than tiling, the latter algorithm can also be used to decide if P can be tiled by 2 × 1 dominos. These are classical problems with important applications in VLSI design, and the related problem of finding a maximum packing of 2 × 2 squares is known to be NP-hard [ 6 ]. For our three problems there are known pseudo-polynomial-time algorithms, that is, algorithms with running times polynomial in the area or perimeter of P . However, the standard, compact way to represent a polygon is by listing the coordinates of the corners in binary. We use this representation, and thus present the first polynomial-time algorithms for the problems. Concretely, we give a simple O(n log n )-time algorithm for tiling with squares, where n is the number of corners of P . We then give a more involved algorithm that reduces the problems of packing and tiling with dominos to finding a maximum and perfect matching in a graph with O ( n 3 ) vertices. This leads to algorithms with running times \(O(n^3 \frac{\log ^3 n}{\log ^2\log n})\) and \(O(n^3 \frac{\log ^2 n}{\log \log n})\) , respectively.

We formulate and optimally solve a new generalized Set Similarity Search problem, which assumes the size of the database and query sets are known in advance. By creating polylog copies of our data-structure, we optimally solve any symmetric Approximate Set Similarity Search problem, including approximate versions of Subset Search, Maximum Inner Product Search (MIPS), Jaccard Similarity Search, and Partial Match. Our algorithm can be seen as a natural generalization of previous work on Set as well as Euclidean Similarity Search, but conceptually it differs by optimally exploiting the information present in the sets as well as their complements, and doing so asymmetrically between queries and stored sets. Doing so we improve upon the best previous work: MinHash [J. Discrete Algorithms 1998], SimHash [STOC 2002], Spherical LSF [SODA 2016, 2017], and Chosen Path [STOC 2017] by as much as a factor n0.14in both time and space; or in the near-constant time regime, in space, by an arbitrarily large polynomial factor. Turning the geometric concept, based on Boolean supermajority functions, into a practical algorithm requires ideas from branching random walks on Z2, for which we give the first non-asymptotic near tight analysis. Our lower bounds follow from new hypercontractive arguments, which can be seen as characterizing the exact family of similarity search problems for which supermajorities are optimal. The optimality holds for among all hashing based data structures in the random setting, and by reductions, for 1 cell and 2 cell probe data structures.

Kernel methods are fundamental tools in machine learning that allow detection of non-linear dependencies between data without explicitly constructing feature vectors in high dimensional spaces. A major disadvantage of kernel methods is their poor scalability: primitives such as kernel PCA or kernel ridge regression generally take prohibitively large quadratic space and (at least) quadratic time, as kernel matrices are usually dense. Some methods for speeding up kernel linear algebra are known, but they all invariably take time exponential in either the dimension of the input point set (e.g., fast multipole methods suffer from the curse of dimensionality) or in the degree of the kernel function. Oblivious sketching has emerged as a powerful approach to speeding up numerical linear algebra over the past decade, but our understanding of oblivious sketching solutions for kernel matrices has remained quite limited, suffering from the aforementioned exponential dependence on input parameters. Our main contribution is a general method for applying sketching solutions developed in numerical linear algebra over the past decade to a tensoring of data points without forming the tensoring explicitly. This leads to the first oblivious sketch for the polynomial kernel with a target dimension that is only polynomially dependent on the degree of the kernel function, as well as the first oblivious sketch for the Gaussian kernel on bounded datasets that does not suffer from an exponential dependence on the dimensionality of input data points.

We show that approximate similarity (near neighbour) search can be solved in high dimensions with performance matching state of the art (data independent) Locality Sensitive Hashing, but with a guarantee of no false negatives. Specifically we give two data structures for common problems. For c-approximate near neighbour in Hamming space, for which we get query time dn^{1/c+o(1)} and space dn^{1+1/c+o(1)} matching that of [Indyk and Motwani, 1998] and answering a long standing open question from [Indyk, 2000a] and [Pagh, 2016] in the affirmative. For (s1, s2)-approximate Jaccard similarity we get query time d^2n^{ρ+o(1)} and space d^2n^{1+ρ+o(1), ρ= [log (1+s1)/(2s1)]/[log (1+s2)/(2s2)], when sets have equal size, matching the performance of [Pagh and Christiani, 2017].We use space partitions as in classic LSH, but construct these using a combination of brute force, tensoring and splitter functions à la [Naor et al., 1995]. We also show two dimensionality reduction lemmas with 1-sided error.

We present a data structure for spherical range reporting on a point set S, i.e., reporting all points in S that lie within radius r of a given query point q (with a small probability of error). Our solution builds upon the Locality-Sensitive Hashing (LSH) framework of Indyk and Motwani, which represents the asymptotically best solutions to near neighbor problems in high dimensions. While traditional LSH data structures have several parameters whose optimal values depend on the distance distribution from q to the points of S (and in particular on the number of points to report), our data structure is essentially parameter-free and only takes as parameter the space the user is willing to allocate. Nevertheless, its expected query time basically matches that of an LSH data structure whose parameters have been optimally chosen for the data and query in question under the given space constraints. In particular, our data structure provides a smooth trade-off between hard queries (typically addressed by standard LSH parameter settings) and easy queries such as those where the number of points to report is a constant fraction of S, or where almost all points in S are far away from the query point. In contrast, known data structures fix LSH parameters based on certain parameters of the input alone. The algorithm has expected query time bounded by O(t(n/t)p), where t is the number of points to report and ρ ∊ (0,1) depends on the data distribution and the strength of the LSH family used. The previously best running time in high dimensions was Ω(tnp), achieved by traditional LSH-based data structures where parameters are tuned for outputting a single point within distance r. Further, for many data distributions where the intrinsic dimensionality of the point set close to q is low, we can give improved upper bounds on the expected query time. We finally present a parameter-free way of using multi-probing, for LSH families that support it, and show that for many such families this approach allows us to get expected query time close to O(np + t), which is the best we can hope to achieve using LSH.

A number of tasks in classification, information retrieval, recommendation systems, and record linkage reduce to the core problem of inner product similarity join (IPS join): identifying pairs of vectors in a collection that have a sufficiently large inner product. IPS join is well understood when vectors are normalized and some approximation of inner products is allowed. However, the general case where vectors may have any length appears much more challenging. Recently, new upper bounds based on asymmetric locality-sensitive hashing (ALSH) and asymmetric embeddings have emerged, but little has been known on the lower bound side. In this paper we initiate a systematic study of inner product similarity join, showing new lower and upper bounds. Our main results are: Approximation hardness of IPS join in subquadratic time, assuming the strong exponential time hypothesis. New upper and lower bounds for (A)LSH-based algorithms. In particular, we show that asymmetry can be avoided by relaxing the LSH definition to only consider the collision probability of distinct elements. A new indexing method for IPS based on linear sketches, implying that our hardness results are not far from being tight.