Related papers: A Generalized Hamming Distance of Sequence Pattern…
Using class labels to represent class similarity is a typical approach to training deep hashing systems for retrieval; samples from the same or different classes take binary 1 or 0 similarity values. This similarity does not model the full…
Base sequences BS(m,n) are quadruples (A;B;C;D) of {+1,-1}-sequences, with A and B of length m and C and D of length n, such that the sum of their nonperiodic autocorrelation functions is a delta-function. Normal sequences NS(n) are base…
Persistent homology allows us to create topological summaries of complex data. In order to analyse these statistically, we need to choose a topological summary and a relevant metric space in which this topological summary exists. While…
Using geometric techniques like projection and dimensionality reduction, we show that there exists a randomized sub-linear time algorithm that can estimate the Hamming distance between two matrices. Consider two matrices ${\bf A}$ and ${\bf…
We develop a systematic method to calculate the trace distance between two reduced density matrices in 1+1 dimensional quantum field theories. The approach exploits the path integral representation of the reduced density matrices and an ad…
We consider the problem of learning a measure of distance among vectors in a feature space and propose a hybrid method that simultaneously learns from similarity ratings assigned to pairs of vectors and class labels assigned to individual…
In this paper we consider the sequence whose n^{th} term is the number of h-vectors of length n. We show that the n^{th} term of this sequence is bounded above by the n^{th} Fibonacci number and bounded below by the number if integer…
In this paper we consider several variants of the pattern matching problem. In particular, we investigate the following problems: 1) Pattern matching with k mismatches; 2) Approximate counting of mismatches; and 3) Pattern matching with…
Given a one-dimensional shift $X$, let $|F_X(\ell)|$ be the number of follower sets of words of length $\ell$ in $X$. We call the sequence $\{|F_X(\ell)|\}_{\ell \in \mathbb{N}}$ the follower set sequence of the shift $X$. Extender sets are…
We consider the expected length of the longest common subsequence between two random words of lengths $n$ and $(1-\varepsilon)kn$ over $k$-symbol alphabet. It is well-known that this quantity is asymptotic to $\gamma_{k,\varepsilon} n$ for…
Quantifying the differences between networks is a challenging and ever-present problem in network science. In recent years a multitude of diverse, ad hoc solutions to this problem have been introduced. Here we propose that simple and…
The Levenshtein distance is an important tool for the comparison of symbolic sequences, with many appearances in genome research, linguistics and other areas. For efficient applications, an approximation by a distance of smaller…
We describe new, simple, recursive methods of construction for orientable sequences over an arbitrary finite alphabet, i.e. periodic sequences in which any sub-sequence of n consecutive elements occurs at most once in a period in either…
Reed-Solomon codes and Gabidulin codes have maximum Hamming distance and maximum rank distance, respectively. A general construction using skew polynomials, called skew Reed-Solomon codes, has already been introduced in the literature. In…
We provide upper and lower bounds for the expected length $\mathbb E(L_{n,m})$ of the longest common pattern contained in $m$ random permutations of length $n$. We also address the tightness of the concentration of $L_{n,m}$ around $\mathbb…
Consider the continuum of points along the edges of a network, i.e., a connected, undirected graph with positive edge weights. We measure the distance between these points in terms of the weighted shortest path distance, called the network…
We redevelop persistent homology (topological persistence) from a categorical point of view. The main objects of study are diagrams, indexed by the poset of real numbers, in some target category. The set of such diagrams has an interleaving…
Orientable sequences of order n are infinite periodic sequences with symbols drawn from a finite alphabet of size k with the property that any particular subsequence of length n occurs at most once in a period in either direction. They were…
Given F:[a,b]^k\to [a,b] and a nonconstant X_0 with P(X_0\in [a,b])=1, define the hierarchical sequence of random variables {X_n}_{n\ge 0} by X_{n+1}=F(X_{n,1},...,X_{n,k}), where X_{n,i} are i.i.d. as X_n. Such sequences arise from…
Sequence comparison is a basic task to capture similarities and differences between two or more sequences of symbols, with countless applications such as in computational biology. An alignment is a way to compare sequences, where a giving…