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http://dx.doi.org/10.1073/pnas.48.7.1134 | DOI Listing |
Neural Netw
January 2025
City University of Hong Kong Shenzhen Research Institute, Shenzhen, China; Department of Mathematics, City University of Hong Kong, Hong Kong, China. Electronic address:
We consider kernel-based supervised learning using random Fourier features, focusing on its statistical error bounds and generalization properties with general loss functions. Beyond the least squares loss, existing results only demonstrate worst-case analysis with rate n and the number of features at least comparable to n, and refined-case analysis where it can achieve almost n rate when the kernel's eigenvalue decay is exponential and the number of features is again at least comparable to n. For the least squares loss, the results are much richer and the optimal rates can be achieved under the source and capacity assumptions, with the number of features smaller than n.
View Article and Find Full Text PDFJ Phys Condens Matter
January 2025
School of Physical Sciences, Indian Association for the Cultivation of Science, 2A & 2B Raja S.C. Mullick Road, Jadavpur, Kolkata, Kolkata, West Bengal, 700032, INDIA.
Periodically driven closed quantum systems are expected to eventually heat up to infinite temperature ; reaching a steady state described by a circular orthogonal ensemble (COE). However, such finite driven systems may exhibit sufficiently long prethermal regimes; their properties in these regimes are qualitatively different from that of their corresponding infinite temperature steady states. These, often experimentally relevant, prethermal regimes host a wide range of phenomena; they may exhibit dynamical localization and freezing, host Floquet scars, display signatures of Hilbert space fragmentation, and exhibit time crystalline phases.
View Article and Find Full Text PDFJ Neural Eng
January 2025
Center for Complex Systems and Brain Sciences, Universidad Nacional de San Martin Escuela de Ciencia Y Tecnologia, 25 de Mayo y Francia, San Martín, Buenos Aires, 1650, ARGENTINA.
Objective Magnetic resonance imaging (MRI), functional MRI (fMRI) and other neuroimaging techniques are routinely used in medical diagnosis, cognitive neuroscience or recently in brain decoding. They produce three- or four-dimensional scans reflecting the geometry of brain tissue or activity, which is highly correlated temporally and spatially. While there exist numerous theoretically guided methods for analyzing correlations in one-dimensional data, they often cannot be readily generalized to the multidimensional geometrically embedded setting.
View Article and Find Full Text PDFEntropy (Basel)
December 2024
Institute of Control & Computation Engineering, University of Zielona Góra, Licealna 9, 65-417 Zielona Góra, Poland.
Infinite-dimensional systems play an important role in the continuous-variable quantum computation model, which can compete with a more standard approach based on qubit and quantum circuit computation models. But, in many cases, the value of entropy unfortunately cannot be easily computed for states originating from an infinite-dimensional Hilbert space. Therefore, in this article, the unified quantum entropy (which extends the standard von Neumann entropy) notion is extended to the case of infinite-dimensional systems by using the Fredholm determinant theory.
View Article and Find Full Text PDFEntropy (Basel)
December 2024
Department of Physics, University of Maryland, College Park, MD 20742-4111, USA.
We define predictive states and predictive complexity for quantum systems composed of distinct subsystems. This complexity is a generalization of entanglement entropy. It is inspired by the statistical or forecasting complexity of predictive state analysis of stochastic and complex systems theory but is intrinsically quantum.
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