Related papers: Compressed Representations of Permutations, and Ap…
We consider a few special cases of the more general question: How many permutations $\pi\in\mathcal{S}_n$ have the property that $\pi^2$ has $j$ descents for some $j$? In this paper, we first enumerate Grassmannian permutations $\pi$ by the…
In computational design and fabrication, neural networks are becoming important surrogates for bulky forward simulations. A long-standing, intertwined question is that of inverse design: how to compute a design that satisfies a desired…
In this article, we develop nested representations for cosine and inverse cosine functions, which is a generalization of Vi\`{e}te's formula for $\pi$. We explore a natural inverse relationship between these representations and develop…
We revisit the problem of permuting an array of length $n$ according to a given permutation in place, that is, using only a small number of bits of extra storage. Fich, Munro and Poblete [FOCS 1990, SICOMP 1995] obtained an elegant…
Many common types of data can be represented as functions that map coordinates to signal values, such as pixel locations to RGB values in the case of an image. Based on this view, data can be compressed by overfitting a compact neural…
Traditional compressed sensing considers sampling a 1D signal. For a multidimensional signal, if reshaped into a vector, the required size of the sensing matrix becomes dramatically large, which increases the storage and computational…
Given a countable set X (usually taken to be the natural numbers or integers), an infinite permutation, \pi, of X is a linear ordering of X. This paper investigates the combinatorial complexity of infinite permutations on the natural…
The compressed indexing problem is to preprocess a string $S$ of length $n$ into a compressed representation that supports pattern matching queries. That is, given a string $P$ of length $m$ report all occurrences of $P$ in $S$. We present…
Compressed sensing is a signal processing method that acquires data directly in a compressed form. This allows one to make less measurements than what was considered necessary to record a signal, enabling faster or more precise measurement…
Reductions combine collections of inputs with an associative (and here, also commutative) operator to produce collections of outputs. When the same value contributes to multiple outputs, there is an opportunity to reuse partial results,…
While learning based compression techniques for images have outperformed traditional methods, they have not been widely adopted in machine learning pipelines. This is largely due to lack of standardization and lack of retention of salient…
A property of sparse representations in relation to their capacity for information storage is discussed. It is shown that this feature can be used for an application that we term Encrypted Image Folding. The proposed procedure is realizable…
We describe an algorithm for splitting permutation representations of finite group over fields of characteristic zero into irreducible components. The algorithm is based on the fact that the components of the invariant inner product in…
A permutation-invariant code on m qubits is a subspace of the symmetric subspace of the m qubits. We derive permutation-invariant codes that can encode an increasing amount of quantum information while suppressing leading order spontaneous…
Recent methods for learning vector space representations of words have succeeded in capturing fine-grained semantic and syntactic regularities using vector arithmetic. However, these vector space representations (created through large-scale…
We generalize the notion of proof term to the realm of transfinite reduction. Proof terms represent reductions in the first-order term format, thereby facilitating their formal analysis. We show that any transfinite reduction can be…
For a word $\pi$ and integer $i$, we define $L^i(\pi)$ to be the length of the longest subsequence of the form $i(i+1)\cdots j$, and we let $L(\pi):=\max_i L^i(\pi)$. In this paper we estimate the expected values of $L^1(\pi)$ and $L(\pi)$…
We investigate compressibility of the dimension of positive semidefinite matrices while approximately preserving their pairwise inner products. This can either be regarded as compression of positive semidefinite factorizations of…
The development of deep learning algorithms has extensively empowered humanity's task automatization capacity. However, the huge improvement in the performance of these models is highly correlated with their increasing level of complexity,…
Compressed indexing is a powerful technique that enables efficient querying over data stored in compressed form, significantly reducing memory usage and often accelerating computation. While extensive progress has been made for…