Related papers: Optimal Lower and Upper Bounds for Representing Se…
We consider a framework for the construction of iterative schemes for operator equations that combine low-rank approximation in tensor formats and adaptive approximation in a basis. Under fairly general assumptions, we obtain a rigorous…
We develop a new technique for proving cell-probe lower bounds for static data structures. Previous lower bounds used a reduction to communication games, which was known not to be tight by counting arguments. We give the first lower bound…
We show that a large fraction of the data-structure lower bounds known today in fact follow by reduction from the communication complexity of lopsided (asymmetric) set disjointness. This includes lower bounds for: * high-dimensional…
This paper is concerned with the development and analysis of an iterative solver for high-dimensional second-order elliptic problems based on subspace-based low-rank tensor formats. Both the subspaces giving rise to low-rank approximations…
Rank/Select dictionaries are data structures for an ordered set $S \subset \{0,1,...,n-1\}$ to compute $\rank(x,S)$ (the number of elements in $S$ which are no greater than $x$), and $\select(i,S)$ (the $i$-th smallest element in $S$),…
We study the problem of representation learning in goal-conditioned hierarchical reinforcement learning. In such hierarchical structures, a higher-level controller solves tasks by iteratively communicating goals which a lower-level policy…
A \emph{covering array} is an $N \times k$ array of elements from a $v$-ary alphabet such that every $N \times t$ subarray contains all $v^t$ tuples from the alphabet of size $t$ at least $\lambda$ times; this is denoted as $\CA_\lambda(N;…
This paper is the continuation of \cite{htl}, where we deal with Lucas sequences. Here we study integers represented by integer sequences which satisfy binary recursive relations. In case of non-degenerate sequences we give bounds for the…
This paper presents new lower and upper bounds for the optimal compression of binary prefix codes in terms of the most probable input symbol, where compression efficiency is determined by the nonlinear codeword length objective of…
We give a simplified and improved lower bound for the simplex range reporting problem. We show that given a set $P$ of $n$ points in $\mathbb{R}^d$, any data structure that uses $S(n)$ space to answer such queries must have…
Complex networks representing social interactions, brain activities, molecular structures have been studied widely to be able to understand and predict their characteristics as graphs. Models and algorithms for these networks are used in…
We consider the problem of optimally compressing and caching data across a communication network. Given the data generated at edge nodes and a routing path, our goal is to determine the optimal data compression ratios and caching decisions…
We consider classification and regression tasks where we have missing data and assume that the (clean) data resides in a low rank subspace. Finding a hidden subspace is known to be computationally hard. Nevertheless, using a non-proper…
Set and multiset variables in constraint programming have typically been represented using subset bounds. However, this is a weak representation that neglects potentially useful information about a set such as its cardinality. For set…
The nonnegative and positive semidefinite (PSD-) ranks are closely connected to the nonnegative and positive semidefinite extension complexities of a polytope, which are the minimal dimensions of linear and SDP programs which represent this…
Tensors are a natural way to express correlations among many physical variables, but storing tensors in a computer naively requires memory which scales exponentially in the rank of the tensor. This is not optimal, as the required memory is…
Tensor train (TT) decomposition provides a space-efficient representation for higher-order tensors. Despite its advantage, we face two crucial limitations when we apply the TT decomposition to machine learning problems: the lack of…
Supporting top-k document retrieval queries on general text databases, that is, finding the k documents where a given pattern occurs most frequently, has become a topic of interest with practical applications. While the problem has been…
We provide new approximation guarantees for greedy low rank matrix estimation under standard assumptions of restricted strong convexity and smoothness. Our novel analysis also uncovers previously unknown connections between the low rank…
We provide new upper and lower bounds on the minimum possible ratio of the spectral and Frobenius norms of a (partially) symmetric tensor. In the particular case of general tensors our result recovers a known upper bound. For symmetric…