A large body of work studies the problem of learning an approximation to an implicit matrix $A\in \mathbb{R}^{m\times n}$ that is only accessible implicitly via matrix-vector product queries (matvec queries) of the form ${x} \rightarrow {A}{x}$ or ${x} \rightarrow {A}^T{x}$. Of particular interest are methods that learn a near-optimal approximation with a fixed sparsity pattern. For example, we might want to learn a near-optimal diagonal, banded, or arrow-head approximation to an implicit matrix $A$. Naturally, the number of matvec queries required to solve this problem depends on the sparsity pattern, which can be encoded as a binary matrix ${S}\in \{0,1\}^{m\times n}$. The query complexity of previous algorithms scales with quantities like the total number of ones in ${S}$, its maximum column/row sparsity, or the chromatic number of a its "conflict graph". These quantities are incomparable: for a given ${S}$, parameterizing by one might yield lower query complexity than another. In this work, we unify and tighten these prior results by providing a nearly sharp characterization of the matvec query complexity of sparse matrix approximation. Generalizing a definition from graph algorithms, let the degeneracy, ${degen}({S})$, denote the smallest number $k$ so that, if we iteratively delete all rows and columns of ${S}$ with $\leq k$ ones, we are left with an empty matrix. We show that a near-optimal approximation to $A$ with sparsity pattern $S$ can be learned with $\tilde{O}({degen}({S}))$ matrix-vector product queries, and $Ω({degen}({S}))$ queries are necessary, for any sparsity pattern ${S}$. Moreover, unlike prior work based on graph coloring, all of our methods run in polynomial time.

Abstract An eight-partition of a finite set of points (respectively, of a continuous mass distribution) in $$\mathbb {R}^3$$ R 3 consists of three planes that divide the space into 8 octants, such that each open octant contains at most 1/8 of the points (respectively, of the mass). In 1966, Hadwiger showed that any mass distribution in $$\mathbb {R}^3$$ R 3 admits an eight-partition; moreover, one can prescribe the normal direction of one of the three planes. The analogous result for finite point sets follows by a standard limit argument. We prove the following variant of this result: any mass distribution (or point set) in $$\mathbb {R}^3$$ R 3 admits an eight-partition for which the intersection of two of the planes is a line with a prescribed direction. Moreover, we present an efficient algorithm for calculating an eight-partition of a set of n points in $$\mathbb {R}^3$$ R 3 (with prescribed normal direction of one of the planes) in time $$O (n^{7/3})$$ O ( n 7 / 3 ) . A preliminary version of this work appeared in SoCG’24 (Aronov et al., 40th International Symposium on Computational Geometry, 2024).

Let $P$ be a polygonal curve with $n$ vertices in the plane. We construct a data structure of size $O(n \log n)$ suited for simplification queries of the following kind. Given a query line $\ell$ and an integer $k\ge1$, find a curve $Q$ on $\ell$ with at most $k$ vertices that minimizes the discrete Fréchet distance to $P$, among all such curves. Using our data structure, a query can be handled in $O(k^2 \log^3 n + k\log^4 n)$ time. More generally, a geometric tree $T$ on $n$ vertices in the plane can be preprocessed into a near-linear-size structure so that, given a pair $u$, $v$ of its vertices, a line $\ell$, and an integer $k\ge1$, one can find a curve $Q$ on $\ell$ with at most $k$ vertices that minimizes the discrete Fréchet distance to the path from $u$ to $v$ in $T$, in time $O(k^2 \mathop{polylog} n)$. For the general dimension-reduction problem, where $P$ is a curve in $\mathbb{R}^d$ ($d \ge 3$), $0 < \varepsilon_0 < 1$ is a real parameter, and a query specifies a $g$-flat $h$ ($1 \le g \le d-1$) and an integer $k \ge 1$, we construct a data structure of size $O(n\log n + f(\varepsilon_0) n)$, where $f(\varepsilon_0)=(1+1/\varepsilon_0)^{(d-1)/2}$, that allows us to find a curve $Q$ on $h$ with at most $k$ vertices, whose discrete Fréchet distance to $P$ is at most $1+\varepsilon_0$ times the distance of $Q^*$ to $P$, where $Q^*$ is such a curve that minimizes the distance to $P$. The query handling time is $O(f(\varepsilon_0) k^2 \log^2 n)$.

An eight-partition of a finite set of points (respectively, of a continuous mass distribution) in ℝ³ consists of three planes that divide the space into 8 octants, such that each open octant contains at most 1/8 of the points (respectively, of the mass). In 1966, Hadwiger showed that any mass distribution in ℝ³ admits an eight-partition; moreover, one can prescribe the normal direction of one of the three planes. The analogous result for finite point sets follows by a standard limit argument. We prove the following variant of this result: Any mass distribution (or point set) in ℝ³ admits an eight-partition for which the intersection of two of the planes is a line with a prescribed direction. Moreover, we present an efficient algorithm for calculating an eight-partition of a set of n points in ℝ³ (with prescribed normal direction of one of the planes) in time O^*(n^{5/2}).

It is unlikely that the discrete Fréchet distance between two curves of length $n$ can be computed in strictly subquadratic time. We thus consider the setting where one of the curves, $P$, is known in advance. In particular, we wish to construct data structures (distance oracles) of near-linear size that support efficient distance queries with respect to $P$ in sublinear time. Since there is evidence that this is impossible for query curves of length $Θ(n^α)$, for any $α> 0$, we focus on query curves of (small) constant length, for which we are able to devise distance oracles with the desired bounds. We extend our tools to handle subcurves of the given curve, and even arbitrary vertex-to-vertex subcurves of a given geometric tree. That is, we construct an oracle that can quickly compute the distance between a short polygonal path (the query) and a path in the preprocessed tree between two query-specified vertices. Moreover, we define a new family of geometric graphs, $t$-local graphs (which strictly contains the family of geometric spanners with constant stretch), for which a similar oracle exists: we can preprocess a graph $G$ in the family, so that, given a query segment and a pair $u,v$ of vertices in $G$, one can quickly compute the smallest discrete Fréchet distance between the segment and any $(u,v)$-path in $G$. The answer is exact, if $t=1$, and approximate if $t>1$.