arXiv subjects

Bernhard G. Bodmann

Publications and source records attributed to Bernhard G. Bodmann.

At least 19 recordsLinked to original sources

arXiv2026

Deconvolution on graphs via linear programming

The main challenge addressed in this paper is to identify individual terms in a superposition of heat kernels on a graph. We establish geometric conditions on the vertices at which these heat kernels are centered and find bounds on the time parameter governing the evolution under the heat semigroup that guarantee a successful recovery. This result can be viewed as a type of deconvolution on a graph. A first main result addresses the setting of a common time parameter for all the heat kernels. We also treat a more general setting when the time parameter depends on the location at which the heat kernel is centered.

math.FA↗

arXiv2024

Coupled Supersymmetry and Ladder Structures Beyond the Harmonic Oscillator

The development of supersymmetric (SUSY) quantum mechanics has shown that some of the insights based on the algebraic properties of ladder operators related to the quantum mechanical harmonic oscillator carry over to the study of more general systems. At this level of generality, pairs of eigenfunctions of so-called partner Hamiltonians are transformed into each other, but the entire spectrum of any one of them cannot be deduced from this intertwining relationship in general -- except in special cases. In this paper, we present a more general structure that provides all eigenvalues for a class of Hamiltonians that do not factor into a pair of operators satisfying canonical commutation relations. Instead of a pair of partner Hamiltonians, we consider two pairs that differ by an overall shift in their spectrum. This is called coupled supersymmetry. In that case, we also develop coherent states and present some uncertainty principles which generalize the Heisenberg uncertainty principle. Coupled SUSY is explicitly realized by an infinite family of differential operators which admit orthonormal bases of eigenfunctions of generalized harmonic oscillators.

math-ph↗

arXiv2022

A scattering transform for graphs based on heat semigroups, with an application for the detection of anomalies in positive time series with underlying periodicities

This paper develops an adaptive version of Mallat's scattering transform for signals on graphs. The main results are norm bounds for the layers of the transform, obtained from a version of a Beurling-Deny inequality that permits to remove the nonlinear steps in the scattering transform. Under statistical assumptions on the input signal, the norm bounds can be refined. The concepts presented here are illustrated with an application to traffic counts which exhibit characteristic daily and weekly periodicities. Anomalous traffic patterns which deviate from these expected periodicities produce a response in the scattering transform.

math.ST↗

arXiv2022

Spikes, Roots, and Modulations: Phase Retrieval for Finitely-Supported Complex Measures

We study the recovery of a finitely supported distribution, a complex linear combination of Dirac measures, from intensity measurements. The distribution $μ=\sum_{j=1}^{s}c_{j}δ_{t_{j}}$ is given by a coefficient vector $c\in\mathbb{C}^s$ and its support $\{t_1, t_2, \dots, t_s\}$ is contained in $ [0,Λ]$ for some $Λ>0$. The intensity measurements evaluate (squared) magnitudes of a set of linear functionals applied to $μ$, obtained by sampling $\hat μ$, the Fourier transform of $μ$, or by evaluating differences between modulated samples. Following a strategy by Alexeev et al., the structure of the linear functionals, and hence of the non-linear magnitude measurement, is encoded with a graph, where the vertices represent point evaluations of $\hat μ$ at $\{v_1, v_2, \dots, v_n\} \subset [-Ω,Ω]$ and each edge represents a (modulated) difference between vertices incident with it. We show that a Ramanujan graph with degree $d \ge 3$ and $n>\frac{6(1 + 6 /\ln(s/ΛΩ)) s}{1-2\sqrt{d-1}/d}$ vertices provides $M=(d+1)n$ magnitudes that are sufficient for identifying the complex measure up to an overall unimodular multiplicative constant. At the cost of including an additional oversampling step and with an additional requirement that $n-1$ is prime, we construct an explicit recovery algorithm that is based on the Prony method.

math.FA↗

arXiv2021

From the Harmonic Oscillator to Time-Frequency Analysis of Chirp Signals

This paper presents a novel approach to understanding the role of harmonic dynamics and gaining a deeper appreciation for its impact within and outside of quantum mechanics. This includes consequences of harmonic dynamics and the uncertainty principle for anomalous diffusion and for the time-frequency analysis of chirp signals. In this approach, we consider a contact transformation to view a system of canonical variables with coordinate $x$ and momentum $p_x$ in the context of a new system of "generalized" coordinates and momentum. This new system is first studied in the context of non-relativistic quantum mechanics. The classical analog is then explored by use of the Poisson bracket equation. From this, new implications are demonstrated in classical phenomena. One is for a new model of Anomalous and Normal Diffusion. In another, we introduce the concept of the "Mixed Fourier Transform" which explores a new Gaussian Fourier Transform kernel in terms of the generalized variables. This has the ultimate objective of "harmonizing" chirp signals or producing a harmonic signal from an otherwise non-harmonic chirp.

quant-ph↗

arXiv2019

Phase retrieval by random binary questions: Which complementary subspace is closer?

Phase retrieval in real or complex Hilbert spaces is the task of recovering a vector, up to an overall unimodular multiplicative constant, from magnitudes of linear measurements. In this paper, we assume that the vector is normalized, but retain only qualitative, binary information about the measured magnitudes by comparing them with a threshold. In more specific, geometric terms, we choose a sequence of subspaces in a real or complex Hilbert space and only record whether a given vector is closer to the subspace than to the complementary subspace. The subspaces have half the dimension of the Hilbert space and are independent, uniformly distributed with respect to the action of the orthogonal or unitary groups. The main goal of this paper is to find a feasible algorithm for approximate recovery based on the information gained about the vector from these binary questions and to establish error bounds for its approximate recovery. We provide a pointwise bound for fixed input vectors and a uniform bound that controls the worst-case scenario among all inputs. Both bounds hold with high probability with respect to the choice of the subspaces. For real or complex vectors of dimension $n$, the pointwise bound requires $m \ge C δ^{-2} n \log(n)$ and the uniform bound $m \ge C δ^{-2} n^2 \log(δ^{-1} n)$ binary questions in order to achieve an accuracy of $δ$. The accuracy $δ$ is measured by the operator norm of the difference between the rank-one orthogonal projections corresponding to the normalized input vector and its approximate recovery.

math.PR↗

arXiv2018

Point Transformations and the Relationships Among Anomalous Diffusion, Normal Diffusion and the Central Limit Theorem

We present new connections among anomalous diffusion (AD), normal diffusion (ND) and the Central Limit Theorem. This is done by defining a point transformation to a new position variable, which we postulate to be Cartesian, motivated by considerations from super-symmetric quantum mechanics. Canonically quantizing in the new position and momentum variables according to Dirac gives rise to generalized negative semi-definite and self-adjoint Laplacian operators. These lead to new generalized Fourier transformations and associated probability distributions, which are form invariant under the corresponding transform. The new Laplacians also lead us to generalized diffusion equations, which imply a connection to the CLT. We show that the derived diffusion equations capture all of the Fractal and Non-Fractal Diffusion equations of O'Shaughnessy and Procaccia. However, we also obtain new equations that cannot (so far as we are able to tell) be expressed as examples of the O'Shaughnessy and Procaccia equations. These equations also possess asymptotics that are related to a CLT but with bi-modal distributions as limits. The results show, in part, that experimentally measuring the diffusion scaling law can determine the point transformation (for monomial point transformations). We also show that AD in the original, physical position is actually ND when viewed in terms of displacements in an appropriately transformed position variable. Finally, we show that there is a new, anomalous diffusion possible for bi-modal probability distributions that also display attractor behavior which is the consequence of an underlying CLT.

math-ph↗

arXiv2017

Binary Parseval frames from group orbits

Binary Parseval frames share many structural properties with real and complex ones. On the other hand, there are subtle differences, for example that the Gramian of a binary Parseval frame is characterized as a symmetric idempotent whose range contains at least one odd vector. Here, we study binary Parseval frames obtained from the orbit of a vector under a group representation, in short, binary Parseval group frames. In this case, the Gramian of the frame is in the algebra generated by the right regular representation. We identify equivalence classes of such Parseval frames with binary functions on the group that satisfy a convolution identity. This allows us to find structural constraints for such frames. We use these constraints to catalogue equivalence classes of binary Parseval frames obtained from group representations. As an application, we study the performance of binary Parseval frames generated with abelian groups for purposes of error correction. We show that $Z_p^q$ is always preferable to $Z_{p^q}$ when searching for best performing codes associated with binary Parseval group frames.

math.RT↗

arXiv2017

On the minimum of the mean-squared error in 2-means clustering

We study the minimum mean-squared error for 2-means clustering when the outcomes of the vector-valued random variable to be clustered are on two touching spheres of unit radius in $n$-dimensional Euclidean space and the underlying probability distribution is the normalized surface measure. For simplicity, we only consider the asymptotics of large sample sizes and replace empirical samples by the probability measure. The concrete question addressed here is whether a minimizer for the mean-squared error identifies the two individual spheres as clusters. Indeed, in dimensions $n \ge 3$, the minimum of the mean-squared error is achieved by a partition that separates the two spheres and has unit distance between the points in each cluster and the respective mean. In dimension $n=2$, however, the minimizer fails to identify the individual spheres; an optimal partition is obtained by a separating hyperplane that does not contain the point at which the spheres touch.

math.PR↗

arXiv2017

From low to high-dimensional moments without magic

We aim to compute the first few moments of a high-dimensional random vector from the first few moments of a number of its low-dimensional projections. To this end, we identify algebraic conditions on the set of low-dimensional projectors that yield explicit reconstruction formulas. We also provide a computational framework, with which suitable projectors can be derived by solving an optimization problem. Finally, we show that randomized projections permit approximate recovery.

math.NA↗

arXiv2017

Low frame coherence via zero-mean tensor embeddings

This paper is concerned with achieving optimal coherence for highly redundant real unit-norm frames. As the redundancy grows, the number of vectors in the frame becomes too large to admit equiangular arrangements. In this case, other geometric optimality criteria need to be identified. To this end, we use an iteration of the embedding technique by Conway, Hardin and Sloane. As a consequence of their work, a quadratic mapping embeds equiangular lines into a simplex in a real Euclidean space. Here, higher degree polynomial maps embed highly redundant unit-norm frames to simplices in high-dimensional Euclidean spaces. We focus on the lowest degree case in which the embedding is quartic.

math.FA↗

arXiv2016

Maximal Orthoplectic Fusion Frames from Mutually Unbiased Bases and Block Designs

The construction of optimal line packings in real or complex Euclidean spaces has shown to be a tantalizingly difficult task, because it includes the problem of finding maximal sets of equiangular lines. In the regime where equiangular lines are not possible, some optimal packings are known, for example, those achieving the orthoplex bound related to maximal sets of mutually unbiased bases. In this paper, we investigate the packing of subspaces instead of lines and determine the implications of maximality in this context. We leverage the existence of real or complex maximal mutually unbiased bases with a combinatorial design strategy in order to find optimal subspace packings that achieve the orthoplex bound. We also show that maximal sets of mutually unbiased bases convert between coordinate projections associated with certain balanced incomplete block designs and Grassmannian 2-designs. Examples of maximal orthoplectic fusion frames already appeared in the works by Shor, Sloane and by Zauner. They are realized in dimensions that are a power of four in the real case or a power of two in the complex case.

math.FA↗

arXiv2016

Fourier and Beyond: Invariance Properties of a Family of Integral Transforms

The Fourier transform is typically seen as closely related to the additive group of real numbers, its characters and its Haar measure. In this paper, we propose an alternative viewpoint; the Fourier transform can be uniquely characterized by an intertwining relation with dilations and by having a Gaussian as an eigenfunction. This broadens the perspective to an entire family of Fourier-like transforms that are uniquely identified by the same dilation property and having Gaussian-like functions as eigenfunctions. We show that these transforms share many properties with the Fourier transform, particularly unitarity, periodicity and eigenvalues. We also establish short-time analogues of these transforms and show a reconstruction property and an orthogonality relation for the short-time transforms.

math.FA↗

arXiv2016

Region-of-Interest reconstruction from truncated cone-beam projections

Region-of-Interest (ROI) tomography aims at reconstructing a region of interest $C$ inside a body using only x-ray projections intersecting $C$ with the goal to reduce overall radiation exposure when only a small specific region of the body needs to be examined. We consider x-ray acquisition from sources located on a smooth curve $Γ$ in $\mathbb{R}^3$ verifying classical Tuy's condition. In this situation, the {\it non-trucated} cone-beam transform $D f$ of smooth densities $f$ admits an explicit inverse $Z$; however $Z$ cannot directly reconstruct $f$ from ROI-truncated projections. To deal with the ROI tomography problem, we introduce a novel reconstruction approach. For densities $f$ in $L^{\infty}(B)$ where $B$ is a bounded ball in $\mathbb{R}^3$, our method iterates an operator $U$ combining ROI-truncated projections, inversion by the operator $Z$ and appropriate regularization operators. Assuming only knowledge of projections corresponding to a spherical ROI $C \subset B$, given $ε>0$, we prove that if $C$ is sufficiently large our iterative reconstruction algorithm converges uniformly to an $ε$-accurate approximation of $f$, where the accuracy depends on the regularity of $f$ quantified in the Sobolev norm $W^5(B)$. This result shows the existence of a critical ROI radius ensuring the convergence of the ROI reconstruction algorithm to $ε$-accurate approximations of $f$. We numerically verified these theoretical results using simulated acquisition of ROI-truncated cone-beam projection data for multiple acquisition geometries. Numerical experiments indicate that the critical ROI radius is fairly small with respect to the support region~$B$.

math-ph↗

arXiv2015

Achieving the orthoplex bound and constructing weighted complex projective 2-designs with Singer sets

Equiangular tight frames are examples of Grassmannian line packings for a Hilbert space. More specifically, according to a bound by Welch, they are minimizers for the maximal magnitude occurring among the inner products of all pairs of vectors in a unit-norm frame. This paper is dedicated to packings in the regime in which the number of frame vectors precludes the existence of equiangular frames. The orthoplex bound then serves as an alternative to infer a geometric structure of optimal designs. We construct frames of unit-norm vectors in $K$-dimensional complex Hilbert spaces that achieve the orthoplex bound. When $K-1$ is a prime power, we obtain a tight frame with $K^2+1$ vectors and when $K$ is a prime power, with $K^2+K-1$ vectors. In addition, we show that these frames form weighted complex projective 2-designs that are useful additions to maximal equiangular tight frames and maximal sets of mutually unbiased bases in quantum state tomography. Our construction is based on Singer's family of difference sets and the related concept of relative difference sets.

math.FA↗

arXiv2015

What is odd about binary Parseval frames?

This paper examines the construction and properties of binary Parseval frames. We address two questions: When does a binary Parseval frame have a complementary Parseval frame? Which binary symmetric idempotent matrices are Gram matrices of binary Parseval frames? In contrast to the case of real or complex Parseval frames, the answer to these questions is not always affirmative. The key to our understanding comes from an algorithm that constructs binary orthonormal sequences that span a given subspace, whenever possible. Special regard is given to binary frames whose Gram matrices are circulants.

math.FA↗

arXiv2014

Algorithms and error bounds for noisy phase retrieval with low-redundancy frames

The main objective of this paper is to find algorithms accompanied by explicit error bounds for phase retrieval from noisy magnitudes of frame coefficients when the underlying frame has a low redundancy. We achieve these goals with frames consisting of $N=6d-3$ vectors spanning a $d$-dimensional complex Hilbert space. The two algorithms we use, phase propagation or the kernel method, are polynomial time in the dimension $d$. To ensure a successful approximate recovery, we assume that the noise is sufficiently small compared to the squared norm of the vector to be recovered. In this regime, the error bound is inverse proportional to the signal-to-noise ratio. Upper and lower bounds on the sample values of trigonometric polynomials are a central technique in our error estimates.

math.FA↗

arXiv2014

Frame potentials and the geometry of frames

This paper concerns the geometric structure of optimizers for frame potentials. We consider finite, real or complex frames and rotation or unitarily invariant potentials, and mostly specialize to Parseval frames, meaning the frame potential to be optimized is a function on the manifold of Gram matrices belonging to finite Parseval frames. Next to the known classes of equal-norm and equiangular Parseval frames, we introduce equidistributed Parseval frames, which are more general than the equiangular type but have more structure than equal-norm ones. We also provide examples where this class coincides with that of Grassmannian frames, the minimizers for the maximal magnitude among inner products between frame vectors. These different types of frames are characterized in relation to the optimization of frame potentials. Based on results by Lojasiewicz, we show that the gradient descent for a real analytic frame potential on the manifold of Gram matrices belonging to Parseval frames always converges to a critical point. We then derive geometric structures associated with the critical points of different choices of frame potentials. The optimal frames for families of such potentials are thus shown to be equal-norm, or additionally equipartitioned, or even equidistributed.

math.FA↗