The Mathematics of Learning: Dealing with Data

By Tomaso A. Poggio, Steve Smale. 2005

569 citations recorded by the source corpus.

Subfields: Artificial Intelligence, Computational Theory and Mathematics

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Citations in the corpus

Lifelong learning of human actions with deep neural network self-organization (2017, background): “Hierarchies provide a convenient trade-off in terms of invariance-selectivity by decomposing a complex task in a hierarchy of simpler ones (Poggio & Smale, 2003).”
Optimal Rates for the Regularized Least-Squares Algorithm (2007, methodology): “In Section 2 we brie∞y recall the main concepts of the regression problem in the context of supervised learning theory, [6],[15],[ 23 ]; however, the formalism could be easily rephrased using the language of non-parametric regression as in [17].”
Deep learning for smart fish farming: applications, opportunities and challenges (2020, background): “Hierarchical architecture learning systems have achieved superior performances in several engineering applications (Poggio & Smale, 2003 ; Mhaskar & Poggio, 2016).”
Calibration, validation, and sensitivity analysis: What's what (2006, background): “One of the recognized mathematical problems of learning theory [60,61] is the nature of the behavior of the choice of generalization of DMð~pÞ (restricting to our context), also called the hypothesis, as the training set increases.”
Some Properties of Regularized Kernel Methods (2004, background): “…minimization problem can also be seen as particular instance of Tikhonov Regularization (Tikhonov and Arsenin, 1977; Mukherjee et al., 2002) for a multivariate function approximation problem which is well known to be ill-posed (Bertero et al., 1988; Evgeniou et al., 2000; Poggio and Smale, 2003).”
Mathematical Methods for Supervised Learning (2005, background): “We shall be interested in the problem of learning an unknown function defined on a set X which takes values in a set Y .”
Learning Real and Boolean Functions: When Is Deep Better Than Shallow (2016, background): “It is equally intriguing to speculate whether in practice (Poggio and Smale, 2003) only sparse functions may be learnable.”
DISCRETIZATION ERROR ANALYSIS FOR TIKHONOV REGULARIZATION (2006, background): “Analysis and Applications, Vol. 4, No. 1 (2006) 81–99 c© World Scientific Publishing Company”
Surrogate-based Analysis and Optimization (2005, background): “(1), it can be shown that independent of the form of the loss function L, the solution of the variational problem has the form (the Representer Theorem; see [23,51]):”
Least squares linear discriminant analysis (2007, methodology): “Regularized least squares classiflcation [ 26 ] is closely related to Regularized Network [5], Least squares SVM [11], and proximal SVMs [9].”
Sympathy for the Details: Dense Trajectories and Hybrid Classification Architectures for Action Recognition (2016, methodology): “Standard iDT features are extracted from those frame-skipped versions and stacked together before descriptor encoding, decreasing the expectation and variance of the condition number [7,50,51] of the extracted feature matrices.”
Shannon sampling and function reconstruction from point values (2004, background): “Taking the functional derivative as in [19] tells us that if c is a minimizer of Q in `(2)(t), then it satisfies (4.”
Learning Sets with Separating Kernels (2012, background): “In recent years, there has been great progress in the development of theory and algorithms for supervised learning, i.e. function approximation problems from random noisy data [9, 21, 27, 50, 68].”
Application of integral operator for regularized least-square regression (2009, background): “The analysis reveals that the essence of this approach is the isomorphism of the square root operator.”