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<h1>Deep networks and bandlimited functions</h1>
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<p><i>March 2, 2019 — Support my next blog post, <a href="https://www.paypal.com/donate/?hosted_button_id=UCLCSJLFL433E">buy me a coffee</a> ☕.</i></p>
<p>In a recent <a href="http://arxiv.org/pdf/1903.00735.pdf">paper</a>, my colleagues and I considered the <a href='http://en.wikipedia.org/wiki/Deep_learning'>deep</a> <a href='http://en.wikipedia.org/wiki/Rectifier_(neural_networks)'>ReLU</a> network approximation of generalized <a href='http://en.wikipedia.org/wiki/Bandlimiting'>bandlimited</a> functions \(f:B=[0,1]^d\rightarrow\mathbb{R}\) of the form
$$
f(\boldsymbol{x}) = \int_{\mathbb{R}^d}F(\boldsymbol{w})K(\boldsymbol{w}\cdot\boldsymbol{x})d\boldsymbol{w},
$$
with \(\mathrm{supp}\,F\subset[-M,M]^d\), \(M\geq1\), and for some <a href='http://en.wikipedia.org/wiki/Square-integrable_function'>square-integrable</a> function \(F:[-M,M]^d\rightarrow\mathbb{C}\) and <a href='http://en.wikipedia.org/wiki/Analytic_function'>analytic</a> \(K:\mathbb{R}\rightarrow\mathbb{C}\).
We showed that, for any <a href='http://en.wikipedia.org/wiki/Measure_(mathematics)'>measure</a> \(\mu\), such functions can be approximated with error \(\epsilon\) in the \(L^2(B,\mu)\)-norm by deep ReLU networks of depth \(L=\mathcal{O}\left(\log_2^2\frac{1}{\epsilon}\right)\) and size \(W=\mathcal{O}\left(\frac{1}{\epsilon^2}\log_2^2\frac{1}{\epsilon}\right)\), up to some constants that depend on \(F\), \(K\), \(\mu\) and \(B\).</p>
<p>Our theorem is based on a result by Maurey, and on the ability of deep ReLU networks to approximate <a href='https://en.wikipedia.org/wiki/Chebyshev_polynomials'>Chebyshev polynomials</a> and <a href='http://en.wikipedia.org/wiki/Analytic_function'>analytic</a> functions efficiently.</p>
<h2>Maurey's theorem</h2>
<p>A famous <a href='http://en.wikipedia.org/wiki/Carath%C3%A9odory%27s_theorem_(convex_hull)'>theorem</a> of <a href='http://en.wikipedia.org/wiki/Constantin_Carath%C3%A9odory'>Carathéodory</a> states that if a point \(x\in\mathbb{R}^d\) lies in the <a href='http://en.wikipedia.org/wiki/Convex_hull'>convex hull</a> of a set \(P\) then \(x\) can be written as the <a href='https://en.wikipedia.org/wiki/Convex_combination'>convex combination</a> of at most \(d+1\) points in \(P\).
Maurey's theorem is an extension of Carathéodory's result to the infinite-dimensional case of <a href='http://en.wikipedia.org/wiki/Hilbert_space'>Hilbert spaces</a>.</p>
<p><b>Theorem (Maurey).</b>
<i>Let \(H\) be a Hilbert space with norm \(\Vert\cdot\Vert\).
Suppose there exists \(G\subset H\) such that for every \(g\in G\), \(\Vert g\Vert\leq b\) for some \(b>0\).
Then, for every \(f\) in the convex hull of \(G\) and every integer \(n\geq 1\), there is a \(f_n\) in the convex hull of \(n\) points in \(G\) and a constant \(c>b^2-\Vert f\Vert^2\) such that \(\Vert f - f_n\Vert^2\leq \frac{c}{n}\).</i></p>
<p>In practice, we used Maurey's theorem to show that there exists
$$
\begin{align}
f_{\epsilon}(\boldsymbol{x}) = \sum_{j=1}^{\lceil 1/\epsilon^2\rceil}b_j\big[\cos(\beta_j)\mathrm{Re}(K(\boldsymbol{w}_j\cdot\boldsymbol{x})) - \sin(\beta_j)\mathrm{Im}(K(\boldsymbol{w}_j\cdot\boldsymbol{x}))\big],
\end{align}
$$
with \(\sum_{j=1}^{\lceil 1/\epsilon^2\rceil}\vert b_j\vert \leq C_F\) and \(\beta_j\in\mathbb{R}\), such that
$$
\Vert f_{\epsilon}(\boldsymbol{x}) - f(\boldsymbol{x})\Vert_{L^2(\mu, B)}
\leq 2C_F\sqrt{\mu(B)}\epsilon.
$$
In other words, the function \(f\) is approximated by a linear combination of analytic functions with error \(2C_F\sqrt{\mu(B)}\epsilon\) in the \(L^2(B,\mu)\)-norm. The task of approximating \(f\) by deep ReLU networks has been reduced to approximating analytic functions by deep networks.</p>
<h2>Chebyshev polynomials and deep ReLU networks</h2>
<p>The <a href='http://en.wikipedia.org/wiki/Chebyshev_polynomials'>Chebyshev polynomials</a> of the first kind play a central role in <a href='http://en.wikipedia.org/wiki/Approximation_theory'>approximation theory</a>. They are defined on \([-1,1]\) via the three-term <a href='http://en.wikipedia.org/wiki/Recurrence_relation'>recurrence relation</a>
$$
T_n(x) = 2xT_{n-1}(x) - T_{n-2}(x), \;\; n\geq 2,
$$
with \(T_0=1\) and \(T_1(x) = x\). We showed that deep ReLU networks can implement the recurrence relation efficiently. Since truncated Chebyshev series can approximate analytic functions exponentially well, so can deep networks.</p>
<hr>
<h4>Blog posts about neural network approximation theory</h4>
<p>2019 <a href="2019-06-25.html">Deep networks and the Kolmogorov–Arnold theorem</a></p>
<p>2019 <a href="2019-03-02.html">Deep networks and bandlimited functions</a></p>
<p>2017 <a href="2017-12-22.html">Deep networks and the curse of dimensionality</a></p>
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