Deep vs. shallow networks : An approximation theory perspective by Mhaskar, Hrushikesh, Poggio, Tomaso

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Deep vs. shallow networks: An approximation theory perspective by Mhaskar, HN, Poggio, T

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On the Routh approximation technique and least squares errors

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An approximation method with rational functions

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Intrinsic nilpotent approximation

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The selection of network functions to approximate prescribed frequency characteristics

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Hierarchical aggregation of linear systems with multiple time scales

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Segmented approximation and analysis of stochastic processes. by Akant, Adnan.

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An Overview of Some Issues in the Theory of Deep Networks by Poggio, Tomaso, Banburski, Andrzej

“…We review our contributions in the areas of approximation theory and optimization. We also introduce a new approach based on cross-validation leave-one-out stability to estimate bounds on the expected error of overparametrized classifiers, such as deep networks. © 2020 Institute of Electrical Engineers of Japan. …”
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Networks and the Best Approximation Property by Girosi, Federico, Poggio, Tomaso

“…From the point of view of approximation theory, however, the property of approximating continous functions arbitrarily well is not sufficient for characterizing good approximation schemes. …”
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On the Convergence of Stochastic Iterative Dynamic Programming Algorithms by Jaakkola, Tommi, Jordan, Michael I., Singh, Satinder P.

“…In this paper we provide a rigorous proof of convergence of these DP-based learning algorithms by relating them to the powerful techniques of stochastic approximation theory via a new convergence theorem. The theorem establishes a general class of convergent algorithms to which both TD(lambda) and Q-learning belong.…”
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Bounds on Urysohn width by Balitskiy, Alexey

“…This notion was introduced in the context of dimension theory, used in approximation theory, appeared in the work of Gromov on systolic geometry, and nowadays it is a metric invariant of independent interest. …”
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An Overview of Some Issues in the Theory of Deep Networks by Poggio, Tomaso, Banburski, Andrzej

“…We review our contributions in the areas of approximation theory and optimization. We also introduce a new approach based on cross-validation leave-one-out stability to estimate bounds on the expected error of overparametrized classifiers, such as deep networks. © 2020 Institute of Electrical Engineers of Japan. …”
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Hierarchically Local Tasks and Deep Convolutional Networks by Deza, Arturo, Liao, Qianli, Banburski, Andrzej, Poggio, Tomaso

“…Recent results in approximation theory have shown that there is an exponential advantage of deep convolutional-like networks in approximating functions with hierarchical locality in their compositional structure. …”
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How Many Subpopulations Is Too Many? Exponential Lower Bounds for Inferring Population Histories by Kim, Younhun, Koehler, Frederic, Moitra, Ankur, Mossel, Elchanan, Ramnarayan, Govind

“…Using a variety of tools from information theory, the theory of extremal polynomials, and approximation theory, we prove new sharp information-theoretic lower bounds on the problem of reconstructing population structure - the history of multiple subpopulations that merge, split, and change sizes over time. …”
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How Many Subpopulations Is Too Many? Exponential Lower Bounds for Inferring Population Histories by Kim, Younhun, Koehler, Frederic, Moitra, Ankur, Mossel, Elchanan, Ramnarayan, Govind

“…Using a variety of tools from information theory, the theory of extremal polynomials, and approximation theory, we prove new sharp information-theoretic lower bounds on the problem of reconstructing population structure - the history of multiple subpopulations that merge, split, and change sizes over time. …”
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How Many Subpopulations Is Too Many?: Exponential Lower Bounds for Inferring Population Histories by Kim, Younhun, Koehler, Frederic, Moitra, Ankur, Mossel, Elchanan, Ramnarayan, Govind

“…Using a variety of tools from information theory, the theory of extremal polynomials, and approximation theory, we prove new sharp information-theoretic lower bounds on the problem of reconstructing population structure—the history of multiple subpopulations that merge, split and change sizes over time. …”
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Theoretical issues in deep networks by Poggio, Tomaso, Banburski, Andrzej, Liao, Qianli

“…We review our recent results toward this goal. In approximation theory both shallow and deep networks are known to approximate any continuous functions at an exponential cost. …”
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Theoretical Issues in Deep Networks by Poggio, Tomaso, Banburski, Andrzej, Liao, Qianli

“…We review our recent results towards this goal. In {\it approximation theory} both shallow and deep networks are known to approximate any continuous functions on a bounded domain at a cost which is exponential (the number of parameters is exponential in the dimensionality of the function). …”
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Accelerating Asymptotically Exact MCMC for Computationally Intensive Models via Local Approximations by Conrad, Patrick R., Marzouk, Youssef M., Pillai, Natesh S., Smith, Aaron

“…Our approach introduces local approximations of these models into the Metropolis-Hastings kernel, borrowing ideas from deterministic approximation theory, optimization, and experimental design. …”
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