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Ben Hayes

Publications and source records attributed to Ben Hayes.

At least 37 records · Page 2Linked to original sources

Approximate homomorphisms and sofic approximations of orbit equivalence relations

We show that for every countable group, any sequence of approximate homomorphisms with values in permutations can be realized as the restriction of a sofic approximation of an orbit equivalence relation. Moreover, this orbit equivalence relation is uniquely determined by the invariant random subgroup of the approximate homomorphism. We record applications of this result to recover various known stability and conjugacy characterizations for almost homomorphisms of amenable groups.

math.GR↗

On the structure of graph product von Neumann algebras

We undertake a comprehensive study of structural properties of graph products of von Neumann algebras equipped with faithful, normal states, as well as properties of the graph products relative to subalgebras coming from induced subgraphs. Among the technical contributions in this paper include a complete bimodule calculation for subalgebras arising from subgraphs. As an application, we obtain a complete classification of when two subalgebras coming from induced subgraphs can be amenable relative to each other. We also give complete characterizations of when the graph product can be full, diffuse, or a factor. Our results are obtained in a broad generality, and we emphasize that they are new even in the tracial setting. They also allow us to deduce new results about when graph products of groups can be amenable relative to each other.

math.OA↗

Strong 1-boundedness, $L^2$-Betti numbers, algebraic soficity, and graph products

We show that graph products of non trivial finite dimensional von Neumann algebras are strongly 1-bounded when the underlying *-algebra has vanishing first L2-Betti number. The proof uses a combination of the following two key ideas to obtain lower bounds on the Fuglede-Kadison determinant of matrix polynomials in a generating set: a notion called ''algebraic soficity'' for *-algebras allowing for the existence of Galois bounded microstates with asymptotically constant diagonals; a probabilistic construction of the authors of permutation models for graph independence over the diagonal.

math.OA↗

Spectral non-concentration near the top for unimodular random graphs

In recent work on equiangular lines, Jiang, Tidor, Yuan, Zhang, and Zhao showed that a connected bounded degree graph has sublinear second eigenvalue multiplicity. More generally they show that there cannot be too many eigenvalues near the top of the spectrum. We extend this result to infinite unimodular random graphs. As a corollary, the spectral distribution of the adjacency operator cannot have an atom at the top. For an infinite regular expander, we deduce that the singularity of the spectral measure at the top satisfies $μ_G[(1-θ)ρ,ρ] \lesssim θ^c$ for some constant $c>0$, where $ρ$ is the spectral radius of the adjacency operator of the graph. This implies new general estimates on the return probabilities of random walks.

math.PR↗

Growth dichotomy for unimodular random rooted trees

We show that the growth of a unimodular random rooted tree $(T,o)$ of degree bounded by $d$ always exists, assuming its upper growth passes the critical threshold $\sqrt{d-1}$. This complements Timar's work who showed the possible nonexistence of growth below this threshold. The proof goes as follows. By Benjamini-Lyons-Schramm, we can realize $(T,o)$ as the cluster of the root for some invariant percolation on the $d$-regular tree. Then we show that for such a percolation, the limiting exponent with which the lazy random walk returns to the cluster of its starting point always exists. We develop a new method to get this, that we call the 2-3-method, as the usual pointwise ergodic theorems do not seem to work here. We then define and prove the Cohen-Grigorchuk co-growth formula to the invariant percolation setting. This establishes and expresses the growth of the cluster from the limiting exponent, assuming we are above the critical threshold.

math.PR↗

AI (r)evolution -- where are we heading? Thoughts about the future of music and sound technologies in the era of deep learning

Artificial Intelligence (AI) technologies such as deep learning are evolving very quickly bringing many changes to our everyday lives. To explore the future impact and potential of AI in the field of music and sound technologies a doctoral day was held between Queen Mary University of London (QMUL, UK) and Sciences et Technologies de la Musique et du Son (STMS, France). Prompt questions about current trends in AI and music were generated by academics from QMUL and STMS. Students from the two institutions then debated these questions. This report presents a summary of the student debates on the topics of: Data, Impact, and the Environment; Responsible Innovation and Creative Practice; Creativity and Bias; and From Tools to the Singularity. The students represent the future generation of AI and music researchers. The academics represent the incumbent establishment. The student debates reported here capture visions, dreams, concerns, uncertainties, and contentious issues for the future of AI and music as the establishment is rightfully challenged by the next generation.

cs.CY↗

A Review of Differentiable Digital Signal Processing for Music & Speech Synthesis

The term "differentiable digital signal processing" describes a family of techniques in which loss function gradients are backpropagated through digital signal processors, facilitating their integration into neural networks. This article surveys the literature on differentiable audio signal processing, focusing on its use in music & speech synthesis. We catalogue applications to tasks including music performance rendering, sound matching, and voice transformation, discussing the motivations for and implications of the use of this methodology. This is accompanied by an overview of digital signal processing operations that have been implemented differentiably. Finally, we highlight open challenges, including optimisation pathologies, robustness to real-world conditions, and design trade-offs, and discuss directions for future research.

cs.SD↗

Vanishing first cohomology and strong 1-boundedness for von Neumann algebras

In this paper, we obtain a new proof result of Shlyakhtenko which states that if $G$ is a sofic, finitely presented group with vanishing first $\ell^2$-Betti number, then $L(G)$ is strongly 1-bounded. Our proof of this result adapts and simplifies Jung's technical arguments which showed strong 1-boundedness under certain conditions on the Fuglede-Kadison determinant of the matrix capturing the relations. Our proof also features a key idea due to Jung which involves an iterative estimate for the covering numbers of microstate spaces. We also provide a short proof using works of Shlyakhtenko and Shalom that the von Neumann algebras of sofic groups with Property T are strongly 1 bounded, which is a special case of another result by the authors.

math.OA↗

The Responsibility Problem in Neural Networks with Unordered Targets

We discuss the discontinuities that arise when mapping unordered objects to neural network outputs of fixed permutation, referred to as the responsibility problem. Prior work has proved the existence of the issue by identifying a single discontinuity. Here, we show that discontinuities under such models are uncountably infinite, motivating further research into neural networks for unordered data.

cs.LG↗

Sinusoidal Frequency Estimation by Gradient Descent

Sinusoidal parameter estimation is a fundamental task in applications from spectral analysis to time-series forecasting. Estimating the sinusoidal frequency parameter by gradient descent is, however, often impossible as the error function is non-convex and densely populated with local minima. The growing family of differentiable signal processing methods has therefore been unable to tune the frequency of oscillatory components, preventing their use in a broad range of applications. This work presents a technique for joint sinusoidal frequency and amplitude estimation using the Wirtinger derivatives of a complex exponential surrogate and any first order gradient-based optimizer, enabling end to-end training of neural network controllers for unconstrained sinusoidal models.

eess.SP↗

Rigid-Body Sound Synthesis with Differentiable Modal Resonators

Physical models of rigid bodies are used for sound synthesis in applications from virtual environments to music production. Traditional methods such as modal synthesis often rely on computationally expensive numerical solvers, while recent deep learning approaches are limited by post-processing of their results. In this work we present a novel end-to-end framework for training a deep neural network to generate modal resonators for a given 2D shape and material, using a bank of differentiable IIR filters. We demonstrate our method on a dataset of synthetic objects, but train our model using an audio-domain objective, paving the way for physically-informed synthesisers to be learned directly from recordings of real-world objects.

cs.SD↗

A random matrix approach to the Peterson-Thom conjecture

The Peterson-Thom conjecture asserts that any diffuse, amenable subalgebra of a free group factor is contained in a unique maximal amenable subalgebra. This conjecture is motivated by related results in Popa's deformation/rigidity theory and Peterson-Thom's results on L^{2}-Betti numbers. We present an approach to this conjecture in terms of so-called strong convergence of random matrices by formulating a conjecture which is a natural generalization of the Haagerup-Thorbjornsen theorem whose validity would imply the Peterson-Thom conjecture. This random matrix conjecture is related to recent work of Collins-Guionnet-Parraud.

math.OA↗

Property (T) and strong $1$-boundedness for von Neumann algebras

The notion of strong 1-boundedness for finite von Neumann algebras was introduced by Jung in arXiv:math/0510576 . This framework provided a free probabilistic approach to study rigidity properties and classification of finite von Neumann algebras. In this paper, we prove that tracial von Neumann algebras with a finite Kazhdan set are strongly 1-bounded. This includes all Property (T) von Neumann algebras with finite dimensional center and group von Neumann algebras of Property (T) groups. This result generalizes all the previous results in this direction due to Voiculescu, Ge, Ge-Shen, Connes-Shlyakhtenko, Jung-Shlyakhtenko, Jung, and Shlyakhtenko. Our proofs are based on analysis of covering estimates of microstate spaces using an iteration technique in the spirit of Jung.

math.OA↗

Neural Waveshaping Synthesis

We present the Neural Waveshaping Unit (NEWT): a novel, lightweight, fully causal approach to neural audio synthesis which operates directly in the waveform domain, with an accompanying optimisation (FastNEWT) for efficient CPU inference. The NEWT uses time-distributed multilayer perceptrons with periodic activations to implicitly learn nonlinear transfer functions that encode the characteristics of a target timbre. Once trained, a NEWT can produce complex timbral evolutions by simple affine transformations of its input and output signals. We paired the NEWT with a differentiable noise synthesiser and reverb and found it capable of generating realistic musical instrument performances with only 260k total model parameters, conditioned on F0 and loudness features. We compared our method to state-of-the-art benchmarks with a multi-stimulus listening test and the Fréchet Audio Distance and found it performed competitively across the tested timbral domains. Our method significantly outperformed the benchmarks in terms of generation speed, and achieved real-time performance on a consumer CPU, both with and without FastNEWT, suggesting it is a viable basis for future creative sound design tools.

cs.SD↗

Relative entropy and the Pinsker product formula for sofic groups

We continue our study of the outer Pinsker factor for probability measure-preserving actions of sofic groups. Using the notion of doubly quenched convergence developed by Austin, we prove that in many cases the outer Pinsker factor of a product action is the product of the outer Pinsker factors. Our results are parallel to those of Seward for Rohklin entropy. We use these Pinsker products formulas to show that for many actions of a sofic group G on X where X is a compact group and the action is by automorphism, the (measurable) outer Pinsker factor of the action of G on X is given as a quotient by a G-invariant, closed, normal subgroup. We use our results to show that if G is sofic and f in M_{n}(Z(G)) is invertible as a convolution operator, then the action of G on the Pontryagin dual of \Z(G)^{\oplus n}/\Z(G)^{\oplus n}f has completely positive measure-theoretic entropy with respect to the Haar measure. This last application requires our previous work connecting topological entropy in the presence as defined by Li-Liang to measure-theoretic entropy in the presence (implicitly defined by Kerr) for actions on compact groups. In particular, we need our previous formulation of measure-theoretic entropy in the presence in terms of a given topological model.

math.DS↗

Harmonic Models and Bernoullicity

We give many examples of algebraic actions which are factors of Bernoulli shifts. These include certain harmonic models over left orderable groups of large enough growth, as well as algebraic actions associated to certain lopsided elements in any left orderable group. For many of our examples, the acting group is amenable so these actions are Bernoulli (and not just a factor of a Bernoulli), but there is no obvious Bernoulli partition.

math.DS↗

A random matrix approach to absorption in free products

This paper gives a free entropy theoretic perspective on amenable absorption results for free products of tracial von Neumann algebras. In particular, we give the first free entropy proof of Popa's famous result that the generator MASA in a free group factor is maximal amenable, and we partially recover Houdayer's results on amenable absorption and Gamma stability. Moreover, we give a unified approach to all these results using $1$-bounded entropy. We show that if $\mathcal{M} = \mathcal{P} * \mathcal{Q}$, then $\mathcal{P}$ absorbs any subalgebra of $\mathcal{M}$ that intersects it diffusely and that has $1$-bounded entropy zero (which includes amenable and property Gamma algebras as well as many others). In fact, for a subalgebra $\mathcal{P} \leq \mathcal{M}$ to have this absorption property, it suffices for $\mathcal{M}$ to admit random matrix models that have exponential concentration of measure and that "simulate" the conditional expectation onto $\mathcal{P}$.

math.OA↗