arXiv subjects

Jameson Cahill

Publications and source records attributed to Jameson Cahill.

At least 19 recordsLinked to original sources

arXiv2026

Approximation theorems in bilipschitz invariant theory

Bilipschitz invariant theory concerns low-distortion embeddings of orbit spaces into Euclidean space. To date, embeddings with the smallest-possible distortion are known for only a few cases, to include: (a) planar rotations, (b) real phase retrieval, and (c) finite reflection groups. Here, we prove that for all three of these cases, the smallest possible distortion is nearly achieved by a composition of a "max filter bank" with a linear transformation. Our proof amounts to a two-step process: first, we show it suffices to demonstrate a certain inclusion of Lipschitz function spaces, and second, we prove that inclusion, using fundamentally different approaches for the three cases. We also show that these cases interact differently with a few related function spaces, which suggests that a unified treatment would be nontrivial.

math.FA↗

arXiv2024

Towards a bilipschitz invariant theory

Consider the quotient of a Hilbert space by a subgroup of its automorphisms. We study whether this orbit space can be embedded into a Hilbert space by a bilipschitz map, and we identify constraints on such embeddings.

math.FA↗

arXiv2022

Group-invariant max filtering

Given a real inner product space $V$ and a group $G$ of linear isometries, we construct a family of $G$-invariant real-valued functions on $V$ that we call max filters. In the case where $V=\mathbb{R}^d$ and $G$ is finite, a suitable max filter bank separates orbits, and is even bilipschitz in the quotient metric. In the case where $V=L^2(\mathbb{R}^d)$ and $G$ is the group of translation operators, a max filter exhibits stability to diffeomorphic distortion like that of the scattering transform introduced by Mallat. We establish that max filters are well suited for various classification tasks, both in theory and in practice.

cs.IT↗

arXiv2020

Lie PCA: Density estimation for symmetric manifolds

We introduce an extension to local principal component analysis for learning symmetric manifolds. In particular, we use a spectral method to approximate the Lie algebra corresponding to the symmetry group of the underlying manifold. We derive the sample complexity of our method for a variety of manifolds before applying it to various data sets for improved density estimation.

cs.LG↗

arXiv2019

Classifying Signals Under a Finite Abelian Group Action: The Finite Dimensional Setting

Let $G$ be a finite group acting on $\mathbb{C}^N$. We study the problem of identifyng the class in $\mathbb{C}^N / G$ of a given signal: this encompasses several types of problems in signal processing. Some instances include certain generalizations of phase retrieval, image recognition, the analysis of textures, etc. In our previous work \cite{prev}, based on an algebraic approach, we constructed a Lipschitz translation invariant transform -- the case when $G$ is cyclic. Here, we extend our results to include all finite Abelian groups. Moreover, we show the existence of a new transform that avoids computing high powers of the moduli of the signal entries--which can be computationally taxing. The new transform does not enjoy the algebraic structure imposed in our earlier work and is thus more flexible. Other (even lower) dimensional representations are explored, however they only provide an almost everywhere (actually generic) recovery which is not a significant drawback for applications. Our constructions are locally robust and provide alternatives to other statistical and neural networks based methods.

math.FA↗

arXiv2019

Optimal Parseval frames: Total coherence and total volume

We introduce three quantities called total coherence, total volume, and nuclear energy, and we show that equiangular Parseval frames maximize all three of these quantities over the set of all Parseval frames. We then show that equiangular Parseval frames also maximize the total volume and nuclear energy over the set of equal norm frames. Along the way we derive a bound on the smallest k-dimensional volume in an equal norm frame which is a generalization of the Welch bound. We conclude with an extensive list of open questions.

math.FA↗

arXiv2019

Complete set of translation invariant measurements with Lipschitz bounds

In image and audio signal classification, a major problem is to build stable representations that are invariant under rigid motions and, more generally, to small diffeomorphisms. Translation invariant representations of signals in $\mathbb{C}^n$ are of particular importance. The existence of such representations is intimately related to classical invariant theory, inverse problems in compressed sensing and deep learning. Despite an impressive body of litereature on the subject, most representations available are either: i) not stable due to the presence of high frequencies; ii) non discriminative; iii) non invariant when projected to finite dimensional subspaces. In the present paper, we construct low dimensional representations of signals in $\mathbb{C}^n$ that are invariant under finite unitary group actions, as a special case we establish the existence of low-dimensional and complete $\mathbb{Z}_m$-invariant representations for any $m\in\mathbb{N}$. Our construction yields a stable, discriminative transform with semi-explicit Lipschitz bounds on the dimension; this is particularly relevant for applications. Using some tools from Algebraic Geometry, we define a high dimensional homogeneous function that is injective. We then exploit the projective character of this embedding and see that the target space can be reduced significantly by using a generic linear transformation. Finally, we introduce the notion of {\it non-parallel} map, which is enjoyed by our function and employ this to construct a Lipschitz modification of it.

math.FA↗

arXiv2017

Constructions of biangular tight frames and their relationships with equiangular tight frames

We study several interesting examples of Biangular Tight Frames (BTFs) - basis-like sets of unit vectors admitting exactly two distinct frame angles (ie, pairwise absolute inner products) - and examine their relationships with Equiangular Tight Frames (ETFs) - basis-like systems which admit exactly one frame angle. We demonstrate a smooth parametrization BTFs, where the corresponding frame angles transform smoothly with the parameter, which "passes through" an ETF answers two questions regarding the rigidity of BTFs. We also develop a general framework of so-called harmonic BTFs and Steiner BTFs - which includes the equiangular cases, surprisingly, the development of this framework leads to a connection with the famous open problem(s) regarding the existence of Mersenne and Fermat primes. Finally, we construct a (chordally) biangular tight set of subspaces (ie, a tight fusion frame) which "Plücker embeds" into an ETF.

math.FA↗

arXiv2016

Phase retrieval in infinite-dimensional Hilbert spaces

The main result of this paper states that phase retrieval in infinite-dimensional Hilbert spaces is never uniformly stable, in sharp contrast to the finite dimensional setting in which phase retrieval is always stable. This leads us to derive stability results for signals depending on how well they are approximated by finite expansions.

math.FA↗

arXiv2016

The gap between the null space property and the restricted isometry property

The null space property (NSP) and the restricted isometry property (RIP) are two properties which have received considerable attention in the compressed sensing literature. As the name suggests, NSP is a property that depends solely on the null space of the measurement procedure and as such, any two matrices which have the same null space will have NSP if either one of them does. On the other hand, RIP is a property of the measurement procedure itself, and given an RIP matrix it is straightforward to construct another matrix with the same null space that is not RIP. %Furthermore, RIP is known to imply NSP and therefore RIP is a strictly stronger assumption than NSP. We say a matrix is RIP-NSP if it has the same null space as an RIP matrix. We show that such matrices can provide robust recovery of compressible signals under Basis pursuit which in many applicable settings is comparable to the guarantee that RIP provides. More importantly, we constructively show that the RIP-NSP is stronger than NSP with the aid of this robust recovery result, which shows that RIP is fundamentally stronger than NSP.

cs.IT↗

arXiv2016

Connectivity and Irreducibility of Algebraic Varieties of Finite Unit Norm Tight Frames

In this paper, we settle a long-standing problem on the connectivity of spaces of finite unit norm tight frames (FUNTFs), essentially affirming a conjecture first appearing in [Dykema and Strawn, 2003]. Our central technique involves continuous liftings of paths from the polytope of eigensteps to spaces of FUNTFs. After demonstrating this connectivity result, we refine our analysis to show that the set of nonsingular points on these spaces is also connected, and we use this result to show that spaces of FUNTFs are irreducible in the algebro-geometric sense, and also that generic FUNTFs are full spark.

math.FA↗

arXiv2015

Phase retrieval

We answer a number of open problems concerning phase retrieval and phase retrieval by projections. In particular, one main theorem classifies phase retrieval by projections via collections of sequences of vectors allowing norm retrieval. Another key result computes the minimal number of vectors needed to add to a frame in order for it to possess the complement property and hence allow phase retrieval. In furthering this idea, in a third main theorem we show that when a collection of subspaces is one subspace short from allowing phase retrieval, then any partition of orthonormal bases from these subspaces into two sets which fail to span, then each spans a hyperplane. We offer many more results in this area as well as provide a large number of examples showing the limitations of the theory.

math.FA↗

arXiv2014

Phase retrieval and norm retrieval

Phase retrieval has become a very active area of research. We will classify when phase retrieval by Parseval frames passes to the Naimark complement and when phase retrieval by projections passes to the orthogonal complements. We introduce a new concept we call norm retrieval and show that this is what is necessary for passing phase retrieval to complements. This leads to a detailed study of norm retrieval and its relationship to phase retrieval. One fundamental result: a frame $\{φ_i\}_{i=1}^M$ yields phase retrieval if and only if $\{Tφ_i\}_{i=1}^M$ yields norm retrieval for every invertible operator $T$.

math.FA↗

arXiv2014

Robust width: A characterization of uniformly stable and robust compressed sensing

Compressed sensing seeks to invert an underdetermined linear system by exploiting additional knowledge of the true solution. Over the last decade, several instances of compressed sensing have been studied for various applications, and for each instance, reconstruction guarantees are available provided the sensing operator satisfies certain sufficient conditions. In this paper, we completely characterize the sensing operators which allow uniformly stable and robust reconstruction by convex optimization for many of these instances. The characterized sensing operators satisfy a new property we call the robust width property, which simultaneously captures notions of widths from approximation theory and of restricted eigenvalues from statistical regression. We provide a geometric interpretation of this property, we discuss its relationship with the restricted isometry property, and we apply techniques from geometric functional analysis to find random matrices which satisfy the property with high probability.

cs.IT↗

arXiv2013

Saving phase: Injectivity and stability for phase retrieval

Recent advances in convex optimization have led to new strides in the phase retrieval problem over finite-dimensional vector spaces. However, certain fundamental questions remain: What sorts of measurement vectors uniquely determine every signal up to a global phase factor, and how many are needed to do so? Furthermore, which measurement ensembles lend stability? This paper presents several results that address each of these questions. We begin by characterizing injectivity, and we identify that the complement property is indeed a necessary condition in the complex case. We then pose a conjecture that 4M-4 generic measurement vectors are both necessary and sufficient for injectivity in M dimensions, and we prove this conjecture in the special cases where M=2,3. Next, we shift our attention to stability, both in the worst and average cases. Here, we characterize worst-case stability in the real case by introducing a numerical version of the complement property. This new property bears some resemblance to the restricted isometry property of compressed sensing and can be used to derive a sharp lower Lipschitz bound on the intensity measurement mapping. Localized frames are shown to lack this property (suggesting instability), whereas Gaussian random measurements are shown to satisfy this property with high probability. We conclude by presenting results that use a stochastic noise model in both the real and complex cases, and we leverage Cramer-Rao lower bounds to identify stability with stronger versions of the injectivity characterizations.

math.FA↗

arXiv2013

Tight and random nonorthogonal fusion frames

First we show that tight nonorthogonal fusion frames a relatively easy to com by. In order to do this we need to establish a classification of how to to wire a self adjoint operator as a product of (nonorthogonal) projection operators. We also discuss the link between nonorthogonal fusion frames and positive operator valued measures, we define and study a nonorthogonal fusion frame potential, and we introduce the idea of random nonorthogonal fusion frames.

math.FA↗

arXiv2013

Phase Retrieval By Projections

The problem of recovering a vector from the absolute values of its inner products against a family of measurement vectors has been well studied in mathematics and engineering. A generalization of this phase retrieval problem also exists in engineering: recovering a vector from measurements consisting of norms of its orthogonal projections onto a family of subspaces. There exist semidefinite programming algorithms to solve this problem, but much remains unknown for this more general case. Can families of subspaces for which such measurements are injective be completely classified? What is the minimal number of subspaces required to have injectivity? How closely does this problem compare to the usual phase retrieval problem with families of measurement vectors? In this paper, we answer or make incremental steps toward these questions. We provide several characterizations of subspaces which yield injective measurements, and through a concrete construction, we prove the surprising result that phase retrieval can be achieved with $2M-1$ projections of arbitrary rank in $\HH_M$. Finally we present several open problems as we discuss issues unique to the phase retrieval problem with subspaces.

math.FA↗

arXiv2013

A note on scalable frames

We study the problem of determining whether a given frame is scalable, and when it is, understanding the set of all possible scalings. We show that for most frames this is a relatively simple task in that the frame is either not scalable or is scalable in a unique way, and to find this scaling we just have to solve a linear system. We also provide some insight into the set of all scalings when there is not a unique scaling. In particular, we show that this set is a convex polytope whose vertices correspond to minimal scalings.

math.FA↗