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Beyond accuracy: generalization properties of bio-plausible temporal credit assignment rules

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arXiv 2022年

作者： Liu, Yuhan Helena Ghosh, Arna Richards, Blake A. Shea-Brown, Eric Lajoie, Guillaume Department of Applied Mathematics University of Washington SeattleWA United States Allen Institute for Brain Science 615 Westlake Ave N SeattleWA United States Computational Neuroscience Center University of Washington SeattleWA United States School of Computer Science McGill University MontrealQC Canada Mila - Quebec AI Institute MontrealQC Canada Department of Neurology and Neurosurgery Montreal Neurological Institute McGill University MontrealQC Canada Canada CIFAR AI Chair CIFAR TorontoON Canada Dept. de Mathématiques et Statistiques Université de Montréal MontrealQC Canada

To unveil how the brain learns, ongoing work seeks biologically-plausible approximations of gradient descent algorithms for training recurrent neural networks (RNNs). Yet, beyond task accuracy, it is unclear if such learning rules converge to solutions that exhibit different levels of generalization than their non-biologically-plausible counterparts. Leveraging results from deep learning theory based on loss landscape curvature, we ask: how do biologically-plausible gradient approximations affect generalization? We first demonstrate that state-of-the-art biologically-plausible learning rules for training RNNs exhibit worse and more variable generalization performance compared to their machine learning counterparts that follow the true gradient more closely. Next, we verify that such generalization performance is correlated significantly with loss landscape curvature, and we show that biologically-plausible learning rules tend to approach high-curvature regions in synaptic weight space. Using tools from dynamical systems, we derive theoretical arguments and present a theorem explaining this phenomenon. This predicts our numerical results, and explains why biologically-plausible rules lead to worse and more variable generalization properties. Finally, we suggest potential remedies that could be used by the brain to mitigate this effect. To our knowledge, our analysis is the first to identify the reason for this generalization gap between artificial and biologically-plausible learning rules, which can help guide future investigations into how the brain learns solutions that generalize. © 2022, CC BY-NC-SA.

关键词： Recurrent neural networks

A discontinuity of the energy of quantum walk in impurities

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arXiv 2021年

作者： Higuchi, Kenta Komatsu, Takashi Konno, Norio Morioka, Hisashi Segawa, Etsuo Department of Mathematical Sciences Ritsumeikan University Noji-Higashi Kusatsu525-8577 Japan Math. Research Institute Calc for Industry Minami Hiroshima732-0816 Japan Department of Applied Mathematics Yokohama National University Hodogaya Yokohama240-8501 Japan Graduate School of Science and Engineering Ehime University Bunkyo-cho 3 Ehime Matsuyama790-8577 Japan Graduate School of Environment Information Sciences Yokohama National University Hodogaya Yokohama240-8501 Japan

We consider the discrete-time quantum walk whose local dynamics is denoted by C at the perturbed region {0,1,·,M-1} and free at the other positions. We obtain the stationary state with a bounded initial state. The initial state is set so that the perturbed region receives the inflow ωnat time n (|ω|=1). From this expression, we compute the scattering on the surface of -1 and M and also compute the quantity how quantum walker accumulates in the perturbed region;namely the energy of the quantum walk, in the long time limit. We find a discontinuity of the energy with respect to the frequency of the inflow. Copyright © 2021, The Authors. All rights reserved.

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Analysis of regularization methods for the solution of ill–posed problems involving discontinuous operators

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arXiv 2020年

作者： Frühauf, F. Scherzer, O. Leitão, A. Department of Computer Science Universität Innsbruck Technikerstraße 25 InnsbruckA-6020 Austria Radon Institute for Comput. Appl. Math. Austrian Academy of Sciences c/o Kepler Universität Linz LinzA-4040 Austria

We consider a regularization concept for the solution of ill–posed operator equations, where the operator is composed of a continuous and a discontinuous operator. A particular application is level set regularization, where we develop a novel concept of minimizers. The proposed level set regularization is capable of handling changing topologies. A functional analytic framework explaining the splitting of topologies is given. The asymptotic limit of the level set regularization method is an evolution process, which is implemented numerically and the quality of the proposed algorithm is demonstrated by solving an inverse source *** Codes 65J20, 47A52 Copyright © 2020, The Authors. All rights reserved.

关键词： Topology

Comparison of amplitude estimation algorithms by implementation

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arXiv 2020年

作者： Yu, Kwangmin Lim, Hyunkyung Rao, Pooja Jin, Dasol Computational Science Initiative Brookhaven National Laboratory UptonNY11973 United States Department of Applied Math. and Stat. Stony Brook University Stony BrookNY11794 United States Department of Mathematics Stony Brook University Stony BrookNY11794 United States

Since the quantum amplitude estimation (QAE) was invented by Brassard et al., 2002, several advanced algorithms have recently been published (Grinko et al., 2019, Aaronson et al, and Suzuki et al., 2020). The main difference between the variants and the original algorithm is that the variants do not need quantum phase estimation (QPE), a key component of the canonical QAE (Brassard et al., 2002), that is composed of many expensive operations on NISQ devices. In this paper, we compare and analyze two of these new QAE approaches (Grinko et al., 2019, and Suzuki et al., 2020) by implementation using the Qiskit package. The comparisons are drawn based on number of oracle queries, quantum circuit depth, and other complexities of implementation for a fixed accuracy. We discuss the strengths and limitations of each algorithm from a computational perspective. Copyright © 2020, The Authors. All rights reserved.

关键词： Quantum theory

Artificial intelligence advances for de novo molecular structure modeling in cryo-EM

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arXiv 2021年

作者： Si, Dong Nakamura, Andrew Tang, Runbang Guan, Haowen Hou, Jie Firozi, Ammaar Cao, Renzhi Hippe, Kyle Zhao, Minglei Division of Computing and Software Systems University of Washington Bothell BothellWA98011 United States Molecular Engineering and Sciences Institute University of Washington Seattle SeattleWA98105 United States Applied and Computational Math Sciences University of Washington Seattle SeattleWA98195 United States Department of Computer Science Saint Louis University Saint. LouisMO63103 United States Department of Computer Science Pacific Lutheran University TacomaWA98447 United States Department of Biochemistry and Molecular Biology University of Chicago ChicagoIL60637 United States

Cryo-electron microscopy (cryo-EM) has become a major experimental technique to determine the structures of large protein complexes and molecular assemblies, as evidenced by the 2017 Nobel Prize. Although cryo-EM has been drastically improved to generate high-resolution three-dimensional (3D) maps that contain detailed structural information about macromolecules, the computational methods for using the data to automatically build structure models are lagging far behind. The traditional cryo-EM model building approach is template-based homology modeling. Manual de novo modeling is very time-consuming when no template model is found in the database. In recent years, de novo cryo-EM modeling using machine learning (ML) and deep learning (DL) has ranked among the top-performing methods in macromolecular structure modeling. Deep-learning-based de novo cryo-EM modeling is an important application of artificial intelligence, with impressive results and great potential for the next generation of molecular biomedicine. Accordingly, we systematically review the representative ML/DL-based de novo cryo-EM modeling methods. And their significances are discussed from both practical and methodological viewpoints. We also briefly describe the background of cryo-EM data processing workflow. Overall, this review provides an introductory guide to modern research on artificial intelligence (AI) for de novo molecular structure modeling and future directions in this emerging field. © 2021, CC BY-NC-ND.

关键词： Electron microscopes

TOPOLOGICAL ENTROPY FOR SHIFTS OF FINITE TYPE OVER AND TREES

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arXiv 2020年

作者： Ban, Jung-Chao Chang, Chih-Hung Hu, Wen-Guei Wu, Yu-Liang Department of Mathematical Sciences National Chengchi University Taipei11605 Taiwan Math. Division National Center for Theoretical Science National Taiwan University Taipei10617 Taiwan Department of Applied Mathematics National University of Kaohsiung Kaohsiung81148 Taiwan College of Mathematics Sichuan University Chengdu610064 China Department of Applied Mathematics National Chiao Tung University Hsinchu30010 Taiwan

We study the topological entropy of hom tree-shifts and show that, although the topological entropy is not a conjugacy invariant for tree-shifts in general, it remains invariant for hom tree higher block shifts. In [16, 17], Petersen and Salama demonstrated the existence of topological entropy for tree-shifts and h(TX) ≥ h(X), where TX is the hom tree-shift derived from X. We characterize a necessary and sufficient condition when the equality holds for the case where X is a shift of finite type. Additionally, two novel phenomena have been revealed for tree-shifts. There is a gap in the set of topological entropy of hom tree-shifts of finite type, making such a set not dense. Last but not least, the topological entropy of a reducible hom tree-shift of finite type can be strictly larger than that of its maximal irreducible *** Codes 37B40 Copyright © 2020, The Authors. All rights reserved.

关键词： Entropy

The square roots of some classical operators

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arXiv 2021年

作者： Mashreghi, Javad Ptak, Marek Ross, William T. Département de Mathématiques et de Statistique Université Laval QuébecQCG1K 0A6 Canada Department of Applied Mathematics University of Agriculture Ul. Balicka 253C Kraków30-198 Poland Department of Mathematics and Computer Science University of Richmond RichmondVA23173 United States

In this paper we give complete descriptions of the set of square roots of certain classical operators, often providing specific formulas. The classical operators included in this discussion are the square of the unilateral shift, the Volterra operator, certain compressed shifts, the unilateral shift plus its adjoint, the Hilbert matrix, and the Cesàro operator. © 2021, CC BY.

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EXPONENTIAL PERIODS AND O-MINIMALITY

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arXiv 2020年

作者： Commelin, Johan Habegger, Philipp Huber, Annette Mathematisch Instituut Utrecht University Budapestlaan 6 Utrecht3584CD Netherlands Department of Mathematics and Computer Science University of Basel Basel4051 Switzerland Math. Institut Universität Freiburg Ernst-Zermelo-Str. 1 Freiburg79102 Germany

Let α ∈ C be an exponential period. This paper shows that the real and imaginary part of α are up to signs volumes of sets definable in the o-minimal structure generated by Q, the real exponential function and sin|[0,1]. This is a weaker analogue of the precise characterisation of ordinary periods as numbers whose real and imaginary part are up to signs volumes of Q-semi-algebraic sets. Furthermore, we define a notion of naive exponential period and compare it to the existing notions using cohomological methods. We compare these definitions with cohomological exponential periods and periods of exponential Nori motives. In particular, naive exponential periods are the same as periods of exponential Nori motives, which justifies that the definition of naive exponential periods singles out the correct set of complex numbers to be called exponential *** Codes 11G35, 14F25, 14F40, 14P10, 03C64 Copyright © 2020, The Authors. All rights reserved.

关键词： Exponential functions

Evolutionary Dynamics of a Lattice Dimer: a Toy Model for Stability vs. Affinity Trade-offs in Proteins

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arXiv 2023年

作者： Loffredo, Emanuele Vesconi, E. Razban, R. Peleg, O. Shakhnovich, E. Cocco, S. Monasson, R. Laboratory of Physics the Ecole Normale Supérieure CNRS UMR 8023 and PSL Research Sorbonne Université 24 rue Lhomond Paris75231 cedex 05 France Laufer Center for Physical and Quantitative Biology Stony Brook University Stony BrookNY United States Computer Science Dept University of Colorado Boulder BoulderCO80309 United States Santa Fe Institute Santa FeNM87501 United States and Ecology & Evolutionary Biology Physics Applied Math Dept University of Colorado Boulder BoulderCO80309 United States Department of Chemistry and Chemical Biology Harvard University CambridgeMA02138 United States

Understanding how a stressor applied on a biological system shapes its evolution is key to achieving targeted evolutionary control. Here we present a toy model of two interacting lattice proteins to quantify the response to the selective pressure defined by the binding energy. We generate sequence data of proteins and study how the sequence and structural properties of dimers are affected by the applied selective pressure, both during the evolutionary process and in the stationary regime. In particular we show that internal contacts of native structures lose strength, while inter-structure contacts are strengthened due to the folding-binding competition. We discuss how dimerization is achieved through enhanced mutability on the interacting faces, and how the designability of each native structure changes upon introduction of the stressor. Copyright © 2023, The Authors. All rights reserved.

关键词： Proteins