Related papers: Cartesian Products Avoiding Patterns
We consider the problem of efficiently solving large-scale linear least squares problems that have one or more linear constraints that must be satisfied exactly. Whilst some classical approaches are theoretically well founded, they can face…
We present a new approach to the problem of enumerating permutations of length n that avoid a fixed consecutive pattern of length m. We use this idea to give explicit upper and lower bounds on the number of permutations avoiding a pattern…
The main motivation for this article is to explore the connections between the existence of certain combinatorial patterns (as in van der Corputs's theorem on arithmetic progressions of length $3$) with well-known tools and theorems for…
We consider the problem of enumerating permutations with exactly r occurrences of the pattern 1324 and derive functional equations for this general case as well as for the pattern avoidance (r=0) case. The functional equations lead to a new…
We consider the conformal decomposition of Einstein's constraint equations introduced by Lichnerowicz and York, on a compact manifold with boundary. We use order relations on appropriate Banach spaces to derive weak solution generalizations…
We consider the task of lossy compression of high-dimensional vectors through quantization. We propose the approach that learns quantization parameters by minimizing the distortion of scalar products and squared distances between pairs of…
This work addresses models (e.g. potential models of directed orbital systems- the manganates) in which an effective reduction dimensionality occurs as a result of a new symmetry which is intermediate between that of global and local gauge…
Empirical models of demand for differentiated products rely on low-dimensional product representations to capture substitution patterns. These representations are increasingly proxied by applying ML methods to high-dimensional, unstructured…
In this paper, we consider pattern avoidance in a subset of words on $\{1,1,2,2,\dots,n,n\}$ called reverse double lists. In particular a reverse double list is a word formed by concatenating a permutation with its reversal. We enumerate…
The gradual patterns that model the complex co-variations of attributes of the form "The more/less X, The more/less Y" play a crucial role in many real world applications where the amount of numerical data to manage is important, this is…
We consider algorithms with access to an unknown matrix $M\in\mathbb{F}^{n \times d}$ via matrix-vector products, namely, the algorithm chooses vectors $\mathbf{v}^1, \ldots, \mathbf{v}^q$, and observes $M\mathbf{v}^1,\ldots,…
The notion of a mesh pattern was introduced recently, but it has already proved to be a useful tool for description purposes related to sets of permutations. In this paper we study eight mesh patterns of small lengths. In particular, we…
Parity functions are fundamental Boolean operations with critical applications across machine learning, cryptography, and error correction. Yet, learning high-dimensional parity functions poses significant challenges: in a general setting,…
We investigate practical algorithms to find or disprove the existence of small subsets of a dataset which, when removed, reverse the sign of a coefficient in an ordinary least squares regression involving that dataset. We empirically study…
Inconsistency in prediction problems occurs when instances that relate in a certain way on condition attributes, do not follow the same relation on the decision attribute. For example, in ordinal classification with monotonicity…
This paper revisits the notion of classical orthogonal polynomials from a broader functional-analytic point of view. It is intended neither as a survey of known results nor as a review of the literature, but rather as a conceptual…
This article carries out a large dimensional analysis of standard regularized discriminant analysis classifiers designed on the assumption that data arise from a Gaussian mixture model with different means and covariances. The analysis…
Modern order and lattice theory provides convenient mathematical tools for pattern mining, in particular for condensed irredundant representations of pattern spaces and their efficient generation. Formal Concept Analysis (FCA) offers a…
Recently, Brualdi and Cao studied $I_k$-avoiding $(0,1)$-matrices by decomposing them into zigzag paths and proved that the maximum number of $1$'s in such a matrix is given by an exact formula. We further study the structure of maximal…
Shrinking target problems in the context of iterated function systems have received an increasing amount of interest in the past few years. The classical shrinking target problem concerns points returning infinitely many times to a sequence…