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Problem 1: Universal Approximation Theorem for Deep Neural Networks with ReLU Activations Statement: Prove that a feedforward neural network with a single hidden layer using ReLU (Rectified Linear Unit) activation functions is a universal approximator. Specifically, show that for any continuous function f : R" → R and for any € > 0, there exists a neural network N with one hidden layer such that: sup f(x) N(x)| < € xЄK where K is a compact subset of Rn. Key Points for the Proof: • • Utilize the properties of ReLU functions to construct piecewise linear approximations of f. Show that the linear combinations of ReLU activations can approximate any continuous function on compact subsets. Address the density of the set of functions representable by the neural network in the space of continuous functions.

Problem 1: Universal Approximation Theorem for Deep Neural Networks with ReLU Activations Statement: Prove that a feedforward neural network with a single hidden layer using ReLU (Rectified Linear Unit) activation functions is a universal approximator. Specifically, show that for any continuous function f : R" → R and for any € > 0, there exists a neural network N with one hidden layer such that: sup f(x) N(x)| < € xЄK where K is a compact subset of Rn. Key Points for the Proof: • • Utilize the properties of ReLU functions to construct piecewise linear approximations of f. Show that the linear combinations of ReLU activations can approximate any continuous function on compact subsets. Address the density of the set of functions representable by the neural network in the space of continuous functions.

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter3: Matrices
Section3.7: Applications
Problem 69EQ
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Problem 1: Universal Approximation Theorem for Deep Neural Networks with ReLU Activations Statement: Prove that a feedforward neural network with a single hidden layer using ReLU (Rectified Linear Unit) activation functions is a universal approximator. Specifically, show that for any continuous function f : R" → R and for any € > 0, there exists a neural network N with one hidden layer such that: sup f(x) N(x)| < € xЄK where K is a compact subset of Rn. Key Points for the Proof: • • Utilize the properties of ReLU functions to construct piecewise linear approximations of f. Show that the linear combinations of ReLU activations can approximate any continuous function on compact subsets. Address the density of the set of functions representable by the neural network in the space of continuous functions.
Transcribed Image Text:Problem 1: Universal Approximation Theorem for Deep Neural Networks with ReLU Activations Statement: Prove that a feedforward neural network with a single hidden layer using ReLU (Rectified Linear Unit) activation functions is a universal approximator. Specifically, show that for any continuous function f : R" → R and for any € > 0, there exists a neural network N with one hidden layer such that: sup f(x) N(x)| < € xЄK where K is a compact subset of Rn. Key Points for the Proof: • • Utilize the properties of ReLU functions to construct piecewise linear approximations of f. Show that the linear combinations of ReLU activations can approximate any continuous function on compact subsets. Address the density of the set of functions representable by the neural network in the space of continuous functions.
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