KMS states of self-similar k-graph C*-algebras

Li, Hui; Yang, Dilian

doi:10.1016/j.jfa.2019.03.003

Cited by 9 publications

(23 citation statements)

References 28 publications

(61 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The weights Ω in the RKHS can be learned using stochastic gradient descent, for a more detailed discussion on the adaptive update procedure with each incoming input sample, please refer to the full KAARMA derivation ( Li and Príncipe, 2016 ). More recently, we have extended Bayesian filtering to the functional space in ( Li and Principe, 2019 ).…”

Section: Methodsmentioning

confidence: 99%

Biologically-Inspired Pulse Signal Processing for Intelligence at the Edge

Li¹,

Prı́ncipe²

2021

Front. Artif. Intell.

View full text Add to dashboard Cite

There is an ever-growing mismatch between the proliferation of data-intensive, power-hungry deep learning solutions in the machine learning (ML) community and the need for agile, portable solutions in resource-constrained devices, particularly for intelligence at the edge. In this paper, we present a fundamentally novel approach that leverages data-driven intelligence with biologically-inspired efficiency. The proposed Sparse Embodiment Neural-Statistical Architecture (SENSA) decomposes the learning task into two distinct phases: a training phase and a hardware embedment phase where prototypes are extracted from the trained network and used to construct fast, sparse embodiment for hardware deployment at the edge. Specifically, we propose the Sparse Pulse Automata via Reproducing Kernel (SPARK) method, which first constructs a learning machine in the form of a dynamical system using energy-efficient spike or pulse trains, commonly used in neuroscience and neuromorphic engineering, then extracts a rule-based solution in the form of automata or lookup tables for rapid deployment in edge computing platforms. We propose to use the theoretically-grounded unifying framework of the Reproducing Kernel Hilbert Space (RKHS) to provide interpretable, nonlinear, and nonparametric solutions, compared to the typical neural network approach. In kernel methods, the explicit representation of the data is of secondary nature, allowing the same algorithm to be used for different data types without altering the learning rules. To showcase SPARK’s capabilities, we carried out the first proof-of-concept demonstration on the task of isolated-word automatic speech recognition (ASR) or keyword spotting, benchmarked on the TI-46 digit corpus. Together, these energy-efficient and resource-conscious techniques will bring advanced machine learning solutions closer to the edge.

show abstract

Section: Methodsmentioning

confidence: 99%

Biologically-Inspired Pulse Signal Processing for Intelligence at the Edge

Li¹,

Prı́ncipe²

2021

Front. Artif. Intell.

View full text Add to dashboard Cite

show abstract

“…The following characterization of aperiodicity for (G, ) can be proved completely similar to [20,Proposition 3.5] and so we skip its proof here. PROPOSITION 3.6.…”

Section: Self-similar P-graph C*-algebrasmentioning

confidence: 99%

“…As[20, Remark 2.4], if (G, ) is a locally faithful P-graph, then Conditions (iv)-(vii) of Definition 3.1 are redundant.…”

mentioning

confidence: 99%

The ideal structures of self-similar -graph C*-algebras

Yang

2020

Ergod. Th. Dynam. Sys.

Self Cite

View full text Add to dashboard Cite

Let $(G,\unicode[STIX]{x1D6EC})$ be a self-similar $k$ -graph with a possibly infinite vertex set $\unicode[STIX]{x1D6EC}^{0}$ . We associate a universal C*-algebra ${\mathcal{O}}_{G,\unicode[STIX]{x1D6EC}}$ to $(G,\unicode[STIX]{x1D6EC})$ . The main purpose of this paper is to investigate the ideal structures of ${\mathcal{O}}_{G,\unicode[STIX]{x1D6EC}}$ . We prove that there exists a one-to-one correspondence between the set of all $G$ -hereditary and $G$ -saturated subsets of $\unicode[STIX]{x1D6EC}^{0}$ and the set of all gauge-invariant and diagonal-invariant ideals of ${\mathcal{O}}_{G,\unicode[STIX]{x1D6EC}}$ . Under some conditions, we characterize all primitive ideals of ${\mathcal{O}}_{G,\unicode[STIX]{x1D6EC}}$ . Moreover, we describe the Jacobson topology of some concrete examples, which includes the C*-algebra of the product of odometers. On the way to our main results, we study self-similar $P$ -graph C*-algebras in depth.

show abstract

“…The successful jump to actions on graphs in [6] and [19] begs the question of whether the theory can be extended to actions on higher-rank graphs. This has been successfully achieved on the group-action side through the work of Li and Yang in [20,21,22], thus giving a higher-rank analogue of [6]. The first main goal of this paper is to build a higher-rank analogue of [19] by constructing a theory of self-similar groupoid actions on higher-rank graphs.…”

Section: Introductionmentioning

confidence: 99%

“…Following the work in [18,19] (see also [4]), as well as [21] for group actions on kgraphs, we devote the majority of this paper to calculating the equilibrium states of our C * -algebras T (G, Λ) and O(G, Λ) under a natural time evolution lifted from a gauge action. Our work generalises the results in [18,19], and it has a number of advantages over the work in [21]. Firstly, we give a description of the equilibrium states of the Nica-Toeplitz algebra, which has a richer KMS structure than its Cuntz-Pimsner quotient.…”

Section: Introductionmentioning

confidence: 99%

C*-algebras of self-similar actions of groupoids on higher-rank graphs and their equilibrium states

Afsar,

Brownlowe,

Ramagge

et al. 2019

Preprint

View full text Add to dashboard Cite

We introduce the notion of a self-similar action of a groupoid G on a finite higher-rank graph. To these actions we associate a compactly aligned product system of Hilbert bimodules, and we show that the corresponding Nica-Toeplitz and Cuntz-Pimsner algebras are universal for generators and relations. We consider natural actions of the real numbers on both algebras and study the KMS states of the associated dynamics. For large inverse temperatures, we describe the simplex of KMS states on the Nica-Toeplitz algebra in terms of traces on the full C * -algebra of G. To study the KMS structure of the Cuntz-Pimsner algebra, we restrict to strongly-connected finite higherrank graphs and a preferred dynamics. We generalise the G-periodicity group of Li and Yang, which is built to encode the periodicity of the underlying graph in the presence of the action of G. We prove that the KMS states of the Cuntz-Pimsner algebra are parametrised by states on the C * -algebra of the G-periodicity group, and we show that if the graph is G-aperiodic and the action satisfies a finite-state condition, then there is at most one KMS state on the Cuntz-Pimsner algebra. We illustrate our results by introducing the notion of a coloured-graph automaton, which we use to construct examples of self-similar actions. We compute the unique KMS states of the Cuntz-Pimsner algebra for some concrete examples.

show abstract

KMS states of self-similar k-graph C*-algebras

Cited by 9 publications

References 28 publications

Biologically-Inspired Pulse Signal Processing for Intelligence at the Edge

Biologically-Inspired Pulse Signal Processing for Intelligence at the Edge

The ideal structures of self-similar -graph C*-algebras

C*-algebras of self-similar actions of groupoids on higher-rank graphs and their equilibrium states

Contact Info

Product

Resources

About