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Error Bounds for a Matrix-Vector Product Approximation with Deep ReLU Neural Networks
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Abstract—Inspired by the depth and breadth of developments on the
theory of deep learning, we pose these fundamental questions: can we
accurately approximate an arbitrary matrix-vector product using deep
rectified linear unit (ReLU) feedforward neural networks (FNNs)? If so,
can we bound the resulting approximation error? Attempting to answer
these questions, we derive error bounds in Lebesgue and Sobolev norms
for a matrix-vector product approximation with deep ReLU FNNs. Since a
matrix-vector product models several problems in wireless communications
and signal processing; network science and graph signal processing; and
network neuroscience and brain physics, we discuss various applications
that are motivated by an accurate matrix-vector product approximation
with deep ReLU FNNs. Toward this end, the derived error bounds offer a
theoretical insight and guarantee in the development of algorithms based
on deep ReLU FNNs.
Title: Error Bounds for a Matrix-Vector Product Approximation with Deep ReLU Neural Networks
Description:
Abstract—Inspired by the depth and breadth of developments on the
theory of deep learning, we pose these fundamental questions: can we
accurately approximate an arbitrary matrix-vector product using deep
rectified linear unit (ReLU) feedforward neural networks (FNNs)? If so,
can we bound the resulting approximation error? Attempting to answer
these questions, we derive error bounds in Lebesgue and Sobolev norms
for a matrix-vector product approximation with deep ReLU FNNs.
Since a
matrix-vector product models several problems in wireless communications
and signal processing; network science and graph signal processing; and
network neuroscience and brain physics, we discuss various applications
that are motivated by an accurate matrix-vector product approximation
with deep ReLU FNNs.
Toward this end, the derived error bounds offer a
theoretical insight and guarantee in the development of algorithms based
on deep ReLU FNNs.
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