Steven B. Damelin
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On Uniform Weighted Deep Polynomial approximation Open
It is a classical result in rational approximation theory that certain non-smooth or singular functions, such as $|x|$ and $x^{1/p}$, can be efficiently approximated using rational functions with root-exponential convergence in terms of de…
Partial transport for point-cloud registration Open
Point cloud registration is an important task in fields like robotics, computer graphics, and medical imaging, involving the determination of spatial relationships between point sets in 3D space. Real-world challenges, such as non-rigid mo…
Subharmonic Kernels, Equilibrium Measures, Energy, and g-Invariance Open
We study Subharmonic Kernels, Equilibrium Measures, Energy, and g-Invariance for a general class of kernels defined on a class of measure metric spaces.
On the Packing Functions of some Linear Sets of Lebesgue Measure Zero Open
We use a characterization of Minkowski measurability to study the asymptotics of best packing on cut-out subsets of the real line with Minkowski dimension $d\in(0,1)$. Our main result is a proof that Minkowski measurability is a sufficient…
Partial Transport for Point-Cloud Registration Open
Point cloud registration plays a crucial role in various fields, including robotics, computer graphics, and medical imaging. This process involves determining spatial relationships between different sets of points, typically within a 3D sp…
The Geometric Approach to the Classification of Signals via a Maximal Set of Signals Open
In this paper we study the scale-space classification of signals via the maximal set of kernels. We use a geometric approach which arises naturally when we consider parameter variations in scale-space. We derive the Fourier transform formu…
A Multiple Parameter Linear Scale-Space for one dimensional Signal Classification Open
In this article we construct a maximal set of kernels for a multi-parameter linear scale-space that allow us to construct trees for classification and recognition of one-dimensional continuous signals similar the Gaussian linear scale-spac…
On best uniform approximation of finite sets by linear combinations of real valued functions using linear programming Open
We study the best approximation problem: \[ \displaystyle \min_{α\in \mathbb R^m}\max_{1\leq i\leq n}\left|y_i -\sum_{j=1}^m α_j Γ_j ({\bf x}_i) \right|. \] Here: $Γ:=\left\{Γ_1,...,Γ_m\right\}$ is a list of functions where for each $1\leq…
A Bounded mean oscillation (BMO) theorem for small distorted diffeomorphisms from $\mathbb R^D$ to $\mathbb R^D$ and PDE Open
This announcement considers the following problem. We produce a bounded mean oscillation theorem for small distorted diffeomorphisms from $\mathbb R^D$ to $\mathbb R^D$. A revision of this announcement is in the memoir preprint: arXiv:2103…
On a realization of motion and similarity group equivalence classes of labeled points in $\mathbb R^k$ with applications to computer vision Open
We study a realization of motion and similarity group equivalence classes of $n\geq 1$ labeled points in $\mathbb R^k,\, k\geq 1$ as a metric space with a computable metric. Our study is motivated by applications in computer vision.
Well Separated Pair Decomposition and power weighted shortest path metric algorithm fusion. Open
For $s$ $>$ 0, we consider an algorithm that computes all $s$-well separated pairs in certain point sets in $\mathbb{R}^{n}$, $n$ $>1$. For an integer $K$ $>1$, we also consider an algorithm that is a permutation of Dijkstra's algorithm, t…
Preprocessing power weighted shortest path data using a s-Well Separated Pair Decomposition Open
For $s$ $>$ 0, we consider an algorithm that computes all $s$-well separated pairs in certain point sets in $\mathbb{R}^{n}$, $n$ $>1$. For an integer $K$ $>1$, we also consider an algorithm that is a permutation of Dijkstra's algorithm, t…
On the Whitney extension problem for near isometries and beyond. Open
In this memoir, we develop a general framework which allows for a simultaneous study of labeled and unlabeled near alignment data problems in $\mathbb R^D$ and the Whitney near isometry extension problem for discrete and non-discrete subse…
On the Whitney near extension problem, BMO, alignment of data, best approximation in algebraic geometry, manifold learning and their beautiful connections: A modern treatment Open
This paper provides fascinating connections between several mathematical problems which lie on the intersection of several mathematics subjects, namely algebraic geometry, approximation theory, complex-harmonic analysis and high dimensiona…
Power weighted shortest paths for clustering Euclidean data Open
We study the use of power weighted shortest path distance functions for clustering high dimensional Euclidean data, under the assumption that the data is drawn from a collection of disjoint low dimensional manifolds. We argue, theoreticall…
A Note on Local Min-Max affine approximations of real-valued convex functions from R^k with applications to computer vision Open
We present a method to find a local Min-Max affine approximant to a given convex function $f:R^k-R$, and show an application to computer vision/graphics.
Analytic and Numerical Analysis of Singular Cauchy integrals with exponential-type weights Open
Let $I=(c,d)$, $c < 0 < d$, $Q\in C^1: I\rightarrow[0,\infty)$ be a function with given regularity behavior on $I$. Write $w:=\exp(-Q)$ on $I$ and assume that $\int_I x^nw^2(x)dx<\infty$ for all $n=0,1,2,\ldots$. For $x\in I$, we consider …
A Note on a Quantitative Form of the Solovay-Kitaev Theorem Open
The problem of finding good approximations of arbitrary 1-qubit gates is identical to that of finding a dense group generated by a universal subset of $SU(2)$ to approximate an arbitrary element of $SU(2)$. The Solovay-Kitaev Theorem is a …
A Koksma-Hlawka-Potential Identity on the $d$ Dimensional Sphere and its Applications to Discrepancy Open
Let $d\geq 2$ be an integer, $S^d\subset {\mathbb R}^{d+1}$ the unit sphere and $σ$ a finite signed measure whose positive and negative parts are supported on $S^d$ with finite energy. In this paper, we derive an error estimate for the qua…
Isometries and Equivalences Between Point Configurations, Extended To\n $\\varepsilon$-diffeomorphisms Open
In this announcement, we deal with the Orthogonal Procrustes Problem, in\nwhich two point configurations are compared in order to construct a map to\noptimally align the two sets. This extends this to\n$\\varepsilon$-diffeomorphisms, intro…
An Algebraic-Coding Equivalence to the Maximum Distance Separable Conjecture Open
In this paper, we provide Algebraic-Coding necessary and sufficient conditions for the Maximum Distance Separable Conjecture to hold.
Isometries and Equivalences Between Point Configurations, Extended To $\varepsilon$-diffeomorphisms Open
In this announcement, we deal with the Orthogonal Procrustes Problem, in which two point configurations are compared in order to construct a map to optimally align the two sets. This extends this to $\varepsilon$-diffeomorphisms, introduce…
On a condition equivalent to the Maximum Distance Separable conjecture Open
We denote by $\mathcal{P}_q$ the vector space of functions from a finite field $\mathbb{F}_q$ to itself, which can be represented as the space $\mathcal{P}_q := \mathbb{F}_q[x]/(x^q-x)$ of polynomial functions. We denote by $\mathcal{O}_n …
On a condition equivalent to the Maximal Distance Separable conjecture. Open
We denote by $\mathcal{P}_q$ the vector space of functions from a finite field $\mathbb{F}_q$ to itself, which can be represented as the space $\mathcal{P}_q := \mathbb{F}_q[x]/(x^q-x)$ of polynomial functions. We denote by $\mathcal{O}_n …
A BMO theorem for $\epsilon$ distorted diffeomorphisms on $\mathbb R^D$ and an application to comparing manifolds of speech and sound Open
This paper deals with a BMO Theorem for $\epsilon$ distorted diffeomorphisms from $\mathbb R^D$ to $\mathbb R^D$ with applications to manifolds of speech and sound.
A BMO theorem for $ε$ distorted diffeomorphisms from $\mathbb R^D$ to $\mathbb R^D$ with applications to manifolds of speech and sound Open
This paper deals with a BMO Theorem for $ε$ distorted diffeomorphisms from $\mathbb R^D$ to $\mathbb R^D$ with applications to manifolds of speech and sound. The material for this paper appears in the research memoir [2].