A Function Space View of Bounded Norm Infinite Width ReLU Nets: The Multivariate Case

Introduction

It has been argued for a while, and is becoming increasingly apparent in recent years, that in terms of complexity control and generalization in neural network training, “the size [magnitude] of the weights is more important then the size [number of weights or parameters] of the network” (Bartlett, 1997; Neyshabur et al., 2014; Zhang et al., 2016). That is, inductive bias and generalization are not achieved by limiting the size of the network, but rather by explicitly (Wei et al., 2019) or implicitly (Nacson et al., 2019; Lyu & Li, 2019) controlling the magnitude of the weights.

In fact, since networks used in practice are often so large that they can fit any function (any labels) over the training data, it is reasonable to think of the network as virtually infinite-sized, and thus able to represent essentially all functions. Training and generalization ability then rests on fitting the training data while controlling, either explicitly or implicitly, the magnitude of the weights. That is, training searches over all functions, but seeks functions with small representational cost, given by the minimal weight norm required to represent the function. This “representational cost of a function” is the actual inductive bias of learning—the quantity that defines our true model class, and the functional we are actually minimizing in order to learn. Understanding learning with overparameterized (virtually infinite) networks thus rests on understanding this “representational cost”, which is the subject of our paper. Representational cost appears to play an important role in generalization performance; indeed Mei & Montanari (2019) show that minimum norm solutions are optimal for generalization in certain simple cases, and recent work on “double descent” curves is an example of this phenomenon (Belkin et al., 2019; Hastie et al., 2019).

We can also think of understanding the representational cost as asking an approximation theory question: what functions can we represent, or approximate, with our de facto model class, namely the class of functions representable with small magnitude weights? There has been much celebrated work studying approximation in terms of the network size, i.e., asking how many units are necessary in order to approximate a target function (Hornik et al., 1989; Cybenko, 1989; Barron, 1993; Pinkus, 1999). But if complexity is actually controlled by the norm of the weights, and thus our true model class is defined by the magnitude of the weights, we should instead ask how large a norm is necessary in order to capture a target function. This revised view of approximation theory should also change how we view issues such as depth separation: rather then asking how increasing depth can reduce the number of units required to fit a function, we should instead ask how increasing depth can reduce the norm required, i.e., how the representational cost we study changes with depth.

where $\mathcal{R}$ is the Radon transform, $\Delta$ is the Laplacian, and $\partial_{b}$ is a partial derivative w.r.t. the offset in the Radon transform (see Section 3 for an explanation of the Radon transform). This characterization is rigorous for odd dimensions $d$ and for functions where the above expressions are classically well-defined (i.e., smooth enough such that all derivatives are finite, and the integrand in the Radon transform is integrable). But for many functions of interest these quantities are not well-defined classically. Instead, in Definition 1, we use duality to rigorously define a semi-norm $\left\|f\right\|_{\mathcal{R}}$ that captures the essence of the above quantities and is well-defined (though possibly infinite) for any $f$ in any dimension. We show that $\left\|f\right\|_{\mathcal{R}}$ precisely captures the representational cost of $f$, and in particular is finite if and only if $f$ can be approximated arbitrarily well by a bounded norm, but possibly unbounded width, ReLU network. Our precise characterization applies to an architecture with unregularized bias terms (as in Savarese et al. (2019)) and a single unregularized linear unit—otherwise a correction accounting for a linear component is necessary, similar but more complex than the term $|f^{\prime}(-\infty)+f^{\prime}(+\infty)|$ in the univariate case, i.e., (1).

As we uncover, the characterization of the representational cost for multivariate functions is unfortunately not as simple as the characterization (1) in the univariate case, where the Radon transform degenerates. Nevertheless, it is often easy to evaluate, and is a powerful tool for studying the representational power of bounded norm ReLU networks. Furthermore, as detailed in Section 5.5, there is no kernel function for which the associated RKHS norm is the same as (2); i.e., training bounded norm neural networks is fundamentally different from kernel learning. In particular, using our characterization we show the following:

We calculate the representational cost of radial “bumps”, and show there are bumps with finite support that have finite representational cost in all dimensions. The representational cost increases as $1/\varepsilon$ for “sharp” bumps of radius $\varepsilon$ (and fixed height). (Section 5.2)

In dimensions greater than one, we show a general piecewise linear function with bounded support has infinite representational cost (i.e., cannot be represented with a bounded norm, even with infinite networks). (Section 5.3)

We obtain a depth separation in terms of norm: we demonstrate a function in two dimensions that is representable using a depth three ReLU network (i.e., with two hidden layers) with small finite norm, but cannot be represented by any bounded-norm depth two (single hidden layer) ReLU network. As far as we are aware, this is the first depth separation result in terms of the norm required for representation. (Section 5.4)

Although the focus of most previous work on approximation theory for neural networks was on the number of units, the norm of the weights was often used as an intermediate step. However, this use does not provide an exact characterization of the representational cost, only a (often very loose) upper bound, and in particular does not allow for depth separation results where a lower bound is needed. See Savarese et al. (2019) for a detailed discussion, e.g., contrasting with the work of Barron (1993; 1994).

The connection between the Radon transform and two-layer neural networks was previously made by Carroll & Dickinson (1989) and Ito (1991), who used it to obtain constructive approximations when studying approximation theory in terms of network size (number of units) for threshold and sigmoidal networks. This connection also forms the foundation of ridgelet transform analysis of functions Candès & Donoho (1999); Candès (1999). More recently, Sonoda & Murata (2017) used ridgelet transform analysis to study the approximation properties of two-layer neural networks with unbounded activation functions, including the ReLU.

While working on this manuscript, we learned through discussions with Matus Telgarsky of his related parallel work. In particular, Telgarsky obtained a calculation formula for the norm required to represent a radial function, paralleling our calculations in Section 5.2, and used it to show that sufficiently smooth radial functions have finite norm in any dimension, and studied how this norm changes with dimension.

Infinite Width ReLU Networks

We repeat here the discussion of Savarese et al. (2019) defining the representational cost of infinite-width ReLU networks, with some corrections and changes that we highlight.

In words, $\overline{R}(f)$ is the minimal limiting representational cost among all sequences of networks converging to $f$ uniformly (while agreeing with $f$ at zero).

We prove in Appendix H that $\overline{R}(f)$ is equivalent to

Hence, learning an unbounded width ReLU network $g_{\theta}$ by fitting some loss functional $L(\cdot)$ while controlling the Euclidean norm of the weights $C(\theta)$ by minimizing

is effectively the same as learning a function $f$ by controlling $\overline{R}(f)$:

Every two-layer ReLU network decomposes into the sum of a network with absolute value units plus a linear partSuch a decomposition follows immediately from the identity $[t]_{+}=\frac{1}{2}(|t|+t)$. As demonstrated by Savarese et al. (2019) in the 1-D setting, the weights on the absolute value units typically determine the representational cost, with a correction term needed if the linear part has large weight. To allow for a cleaner formulation of the representation cost without this correction term, we consider adding in one additional unregularized linear unit ${\bm{v}}^{\top}{\bm{x}}$ (similar to a “skip connection”) to “absorb” any representational cost due to the linear part.

In fact, we show the minimizer of (13) is unique and is characterized as follows:

The proof of Lemma 1 is given in Appendix I. The uniqueness in Lemma 1 allows for a more explicit characterization $\overline{R}_{1}(f)$ in function space relative to $\overline{R}(f)$, as we show in Section 4.

The Radon transform and its dual

Our characterization of the representational cost $\overline{R}_{1}(f)$ in Section 4 is posed in terms of the Radon transform — a transform that is fundamental to computational imaging, and whose inverse is the basis of image reconstruction in computed tomography. For an investigation of its properties and applications, see Helgason (1999). Here we give a brief review of the Radon transform and its dual as needed for subsequent derivations; readers familiar with these topics can skip to Section 4.

where $\gamma_{d}=\frac{1}{2(2\pi)^{d-1}}$.

where fractional powers of $-\partial^{2}_{b}$ can be defined in Fourier domain, same as fractional powers of the Laplacian. In particular, if $d$ is odd, $(-\partial^{2}_{b})^{(d-1)/2}=(-1)^{(d-1)/2}\partial_{b}^{d-1}$, while if $d$ is even, $(-\partial^{2}_{b})^{(d-1)/2}=(\mathcal{H}\partial_{b})^{d-1}$ where $\mathcal{H}$ is the Hilbert transform in the offset variable $b$.

Representational cost in function space: the ℛℛ\mathcal{R}-norm

Differentiating twice inside the integral, the Laplacian $\Delta f({\bm{x}})=\sum_{i=1}^{d}\partial_{x_{i}}^{2}f({\bm{x}})$ is given by

where $\delta(\cdot)$ denotes a Dirac delta. We see that the right-hand side of (22) is precisely the dual Radon transform of $\alpha$, i.e., we have shown $\Delta f=\mathcal{R}^{*}\{\alpha\}$. Applying the inversion formula for the dual Radon transform given in (17) to this identity, and using the characterization of $\overline{R}_{1}(f)$ given in Lemma 1, immediately gives the following result.

See Figure 2 for an illustration of Lemma 3 in the case $d=2$. This result suggests that more generally if we are given a function $f$, we ought to be able to compute $\overline{R}_{1}(f)$ using the formula in Lemma 3. The following result, proved in Appendix I, shows this is indeed the case assuming $f$ is integrable and sufficiently smooth, which for simplicity we state in the case of odd dimensions $d$. For $d$ even, Proposition 1 holds with the pseudo-differential operators $(-\Delta)^{(d+1)/2}$ and $(-\partial_{b}^{2})^{(d+1)/2}$ in place of $\Delta^{(d+1)/2}$ and $\partial_{b}^{d+1}$; see Section 3..

Here we used the intertwining property of the Radon transform and the Laplacian to write $\mathcal{R}\{\Delta^{(d+1)/2}f\}=\partial_{b}^{d+1}\mathcal{R}\{f\}$ (see Section 3 for more details).

Given these results, one might expect for an arbitrary function $f$ we should have $\overline{R}_{1}(f)$ equal to one of the expressions in (23). However, for many functions of interest these quantities are not classically well-defined. For example, the finite-width ReLU net $f({\bm{x}})=\sum_{i=1}^{n}a_{i}[{\bm{w}}_{i}^{\top}{\bm{x}}-b_{i}]_{+}$ is a piecewise linear function that is non-smooth along each hyperplane ${\bm{w}}_{i}^{\top}{\bm{x}}=b_{i}$, so its derivatives can only be understood in the sense of generalized functions or distributions. Similarly, in this case the Radon transform of $f$ is not well-defined since $f$ is unbounded and not integrable along hyperplanes.

then restrict $\psi$ to a space where $\Delta^{(d+1)/2}\mathcal{R}^{*}\{\psi\}$ is always well-defined. More formally, we have:

We prove in Appendix I that the $\mathcal{R}$-norm is well-defined, though not always finite, for all Lipschitz functions and, whether finite or infinite, is always equal to the representational cost $\overline{R}_{1}(\cdot)$:

$\overline{R}_{1}(f)=\left\|f\right\|_{\mathcal{R}}$ for all functions $f$. In particular, $\overline{R}_{1}(f)$ is finite if and only if $f$ is Lipschitz and $\left\|f\right\|_{\mathcal{R}}$ is finite.

We give the proof of Theorem 1 in Appendix I, but the following example illustrates many key elements of the proof.

Letting $\psi^{*}$ be any even Schwartz function such that $\psi^{*}({\bm{w}}_{i},b_{i})=\psi^{*}(-{\bm{w}}_{i},-b_{i})=\text{sign}(a_{i})$ for all $i=1,...,k$ and $|\psi^{*}({\bm{w}},b)|\leq 1$ otherwise, we see that $\overline{R}_{1}(f)=\left\|f\right\|_{\mathcal{R}}=\sum_{i=1}^{k}|a_{i}|$.

The representational cost $\overline{R}(f)$ defined without the unregularized linear unit is more difficult to characterize explicitly. However, we prove that $\overline{R}(f)$ is finite if and only if $\left\|f\right\|_{\mathcal{R}}$ is finite, and give bounds for $\overline{R}(f)$ in terms of $\left\|f\right\|_{\mathcal{R}}$ and the norm of the gradient of the function “at infinity”, similar to the expressions derived in Savarese et al. (2019) in the 1-D setting.

$\overline{R}(f)$ is finite if and only if $\left\|f\right\|_{\mathcal{R}}$ is finite, in which case we have the bounds

We give the proof of Theorem 2 in Appendix J. The lower bound $\max\{\left\|f\right\|_{\mathcal{R}},2\|\nabla f(\infty)\|\}$ is analogous to the expression for the 1D representational cost (1) obtained in Savarese et al. (2019). From this, one might speculate that $\overline{R}(f)$ is equal to $\max\{\left\|f\right\|_{\mathcal{R}},2\|\nabla f(\infty)\|_{2}\}$. However, in Appendix J we show this is not the case: there are examples of functions $f$ in all dimensions such that $\overline{R}(f)$ attains the upper bound in a non-trivial way (e.g., $f(x,y)=|x|+y$ in $d=2$).

In Appendix K we prove several useful properties for the $\mathcal{R}$-norm. In particular, we show the $\mathcal{R}$-norm is in fact a semi-norm, i.e., it is absolutely homogeneous and satisfies the triangle inequality, while $\left\|f\right\|_{\mathcal{R}}=0$ if and only if $f$ is affine. We also show $\mathcal{R}$-norm is invariant to coordinate translation and rotations, and prove the following scaling law under contractions/dilation:

If $f_{\varepsilon}({\bm{x}}):=f({\bm{x}}/\varepsilon)$ for any $\varepsilon>0$, then $\left\|f_{\varepsilon}\right\|_{\mathcal{R}}=\varepsilon^{-1}\left\|f\right\|_{\mathcal{R}}$

Proposition 2 shows that “spikey” functions will necessarily have large $\mathcal{R}$-norm. For example, let $f$ be any non-negative function supported on the ball of radius 1 with maximum height 1 such that $\left\|f\right\|_{\mathcal{R}}$ is finite. Then the contraction $f_{\varepsilon}$ is supported on the ball of radius $\varepsilon$ with maximum height 1, but $\left\|f_{\varepsilon}\right\|_{\mathcal{R}}=\varepsilon^{-1}\left\|f\right\|_{\mathcal{R}}$ blows up as $\varepsilon\rightarrow 0$.

From a generalization perspective, the fact that the $\mathcal{R}$-norm blows up with contractions is a desirable property, since otherwise the minimum norm fit to data would be spikes on data points. In particular, this is what would happen if the representational cost involved derivatives lower than $d+1$, and so in this sense it is not a coincidence that $\left\|f\right\|_{\mathcal{R}}$ involves derivatives of order $d+1$.

Finally, we show the smoothness requirements of the $\mathcal{R}$-norm are also reflected in Fourier domain. In particular, we show that for a broad class of functions in order $\mathcal{R}$-norm to be finite the Fourier transform of $f$ must decay rapidly along every ray. A precise statement is given in Proposition 12 in Appendix K.

Consequences, Applications and Discussion

Our characterization of the representational cost for multivariate functions in terms of the $\mathcal{R}$-norm is unfortunately not as simple as the characterization in the univariate case. Nevertheless, it is often easy to evaluate, and is a powerful tool for studying the representational power of bounded norm ReLU networks.

Here we relate Sobolev spaces and the $\mathcal{R}$-norm. The key result is the following upper bound, which is proved in Appendix L.

Recall that if the dimension $d$ is odd then $(-\Delta)^{(d+1)/2}$ is just an integer power of the negative Laplacian, which is a linear combination of partial derivatives of order $d+1$. Hence, we have $\|(-\Delta)^{(d+1)/2}f\|_{1}\leq c_{d}\gamma_{d}\|f\|_{W^{d+1,1}}$, where $\|f\|_{W^{d+1,1}}$ is the Sobolev norm given by the sum of $L^{1}$-norm of $f$ and the $L^{1}$-norms of all its weak partial derivatives up to order $d+1$. This gives the following immediate corollary to Proposition 3:

Corollary 1 shows that the space of functions with finite $\mathcal{R}$-norm is “dense” in the space of all functions, in the sense that it contains a full Sobolev space.

2 Radial bump functions

where $\rho(b)=\int_{b}^{\infty}g(t)(t^{2}-b^{2})^{(d-3)/2}t\,dt$,

For example, in the $d=3$ dimensional case, we have

More generally, for any odd dimension $d\geq 3$ a simple induction shows (32) is equivalent to

where $Q_{d}$ is a differential operator of degree $(d+3)/2$ having the form $Q_{d}=\sum_{k=2}^{(d+3)/2}p_{k,d}(b)\partial^{k}$ where each $p_{k,d}(b)$ is a polynomial in $b$ of degree $k-2$. In particular, if the weak derivative $\partial^{(d+1)/2}g$ exists and has bounded variation, then $\left\|f\right\|_{\mathcal{R}}$ is finite.

which is non-negative, supported on the unit ball, and has maximum height $f(\bm{0})=1$, and let $f_{\varepsilon}({\bm{x}})=f({\bm{x}}/\varepsilon)$ be the contraction of $f$ to a ball of radius $\varepsilon$ with the same height. Then using formula (33), and the dilation property (2), we can compute

Note that if we move up to dimension $d=5$, then the function defined by (35) no longer has finite norm since its derivatives of order $(d+3)/2=4$ do not exist; this phenomenon is explored in more detail in the next example.

for any $k>0$. We prove $\|f_{d,k}\|_{\mathcal{R}}$ is finite if and only if $k\geq\frac{d+1}{2}$ (see Appendix M). To illustrate the scaling with dimension $d$, in Appendix M we prove that for the choice $k_{d}=(d+1)/2+2$ we have the bounds $(d+5)d\leq\left\|f_{d,k_{d}}\right\|_{\mathcal{R}}\leq 2d(d+5)$, hence $\left\|f_{d,k_{d}}\right\|_{\mathcal{R}}\sim d^{2}$. Similarly, by the dilation property (2), a contraction of $f_{d,k_{d}}$ to the ball of radius $\varepsilon$ will have $\mathcal{R}$-norm scaling as $\sim d^{2}/\varepsilon$.

The next exampleThe existence of such a radial function was noted in parallel work by Matus Telgarsky. Discussions with Telgarsky motivated us to construct and analyze it using the $\mathcal{R}$-norm. shows there there is a universal choice of radial bump function in all (odd) dimensions with finite $\mathcal{R}$-norm:

Since $g$ is $C^{\infty}$-smooth and its derivatives of all orders are $L^{1}$-bounded, $f$ has finite $\mathcal{R}$-norm by Proposition 4.

3 Piecewise Linear functions

Every finite-width two-layer ReLU network is a continuous piecewise linear function. However, the reverse is not true. For example, in dimensions two and above no compactly supported piecewise linear function is expressible as a finite-width two-layer ReLU network. A natural question then is: what piecewise linear functions are represented by bounded norm infinite-width nets, i.e., have finite $\mathcal{R}$-norm? In particular, can a compactly supported piecewise linear function have finite $\mathcal{R}$-norm? Here we show this is generally not the case.

Before stating our result, we will need a few definitions relating to the geometry of piecewise linear functions. Recall that any piecewise linear function (with finitely many pieces) is divided into polyhedral regions separately by a finite number of boundaries. Each boundary is $(d-1)$-dimensional and contained in a unique hyperplane. Hence, with every boundary we associate the unique (up to sign) unit normal to the hyperplane containing it, which we call the boundary normal. Additionally, in the case of compactly supported piecewise linear function, every boundary set that touches the complement of the support set we call an outer boundary, otherwise we call it an inner boundary.

The following result is proved in Appendix N, and is a consequence of the Fourier decay estimates established in Appendix K.

at least one of the boundary normals is not parallel with every other boundary normal, or

$f$ is everywhere convex (or everywhere concave) when restricted to its support, and at least one of the inner boundary normals is not parallel with all outer boundary normals.

Then $f$ has infinite $\mathcal{R}$-norm.

Note that condition (a) holds for a “generic” piecewise linear function with compact support, i.e., if a function fails to satisfy (a) we can always perturb it slightly such that (a) holds. In this sense no “generic” compactly supported piecewise linear function has finite $\mathcal{R}$-norm. In fact, we are not aware of any compactly supported piecewise linear function with finite $\mathcal{R}$-norm, but our theory does not rule them out a priori.

This result suggests that the space of piecewise linear functions expressible as a bounded norm infinite-width two-layer ReLU network is not qualitatively different than those captured by finite-width networks. We go further and make the following conjecture:

A continuous piecewise linear function $f$ has finite $\mathcal{R}$-norm if and only if it is exactly representable by a finite-width two-layer ReLU network.

4 Depth Separation

In an effort to understand the power of deeper networks, there has been much work showing how some functions can be much more easily approximated in terms of number of required units by deeper networks compared to shallower ones, including results showing how functions that can be well-approximated by three-layer networks require a much larger number of units to approximate if using a two-layer network (e.g. Pinkus (1999); Telgarsky (2016); Liang & Srikant (2016); Safran & Shamir (2017); Yarotsky (2017)). The following example shows that, also in terms of the norm, such a depth separation exists for ReLU nets:

The pyramid function $f({\bm{x}})=[1-\|{\bm{x}}\|_{1}]_{+}$ is a compactly supported piecewise linear function that satisfies condition $(b)$ of Proposition 5, hence has infinite representational cost as a two-layer ReLU network ($\overline{R}(f)=\overline{R}_{1}(f)=+\infty$), but can be exactly represented as a finite-width three-layer ReLU network.

Interestingly, this result shows that, in terms of the norm, we have a qualitative rather then quantitative depth separation: the required norm with three layers is finite, while with only two layers it is not merely very large, but infinite. In contrast, in standard depth separation results, the separation is quantitative: we can compensate for a decrease in depth and use more neurons to achieve the same approximation quality. It would be interesting to further strengthen Example 5 by obtaining a quantitative lower bound on the norm required to $\epsilon$-approximate the pyramid with an infinite-width two-layer ReLU network.

5 The ℛℛ\mathcal{R}-norm is not a RKHS norm

There is an ongoing debate in the community on whether neural network learning can be simulated or replicated by kernel machines with the “right” kernel. In this context, it is interesting to ask whether the inductive bias we uncover can be captured by a kernel, or in other words whether the $\mathcal{R}$-norm is an RKHS (semi-)norm. The answer is no:

The $\mathcal{R}$-norm is not a RKHS (semi-)norm.

6 Generalization implications

We are grateful to Matus Telgarsky (University of Illinois, Urbana-Champaign) for stimulating discussions, including discussing his yet unpublished work with us. In particular, Telgarsky helped us refine our view of radial bumps and realize a fixed radial function can have finite norm in all dimensions. We would also like to thank Guillaume Bal (University of Chicago) for helpful discussions regarding the Radon transform, and Jason Altschuler (MIT) for pointers regarding convergence of measures and Prokhorov’s Theorem. Some of the work was done while DS and NS were visiting the Simons Institute for Theoretical Computer Science as participants in the Foundations of Deep Learning Program. NS was partially supported by NSF awards 1764032 and 1546500. DS was partially supported by the Israel Science Foundation (grant No. 31/1031), and by the Taub Foundation. RW and GO were partially supported by AFOSR FA9550‐18‐1‐0166, NSF IIS‐1447449, NSF DMS‐1930049, and DMS‐1925101.

References

Appendices

Likewise, by the identity $[t]_{+}-[-t]_{+}=t$ we have

We will need the following fact about even and odd decompositions of measures under the total variation norm:

A similar argument shows $\|\alpha^{-}\|_{1}\leq\|\alpha\|_{1}$. ∎

which shows $h_{\alpha}$ is Lipschitz with $\|h_{\alpha}\|_{L}\leq\|\alpha\|_{1}/2$.

H Optimization characterization of representational cost

Here we establish the optimization equivalents of the representational costs $\overline{R}(f)$ and $\overline{R}_{1}(f)$ given in (9) and $\eqref{eq:opt1}$.

As an intermediate step, we first give equivalent expressions for $\overline{R}(f)$ and $\overline{R}_{1}(f)$ in terms of sequences finite-width two-layer ReLU networks converging pointwise to $f$. For this we need to introduce some additional notation and definitions.

Now we establish the following equivalent expressions for the representational costs $\overline{R}(\cdot)$ and $\overline{R}_{1}(\cdot)$.

We prove the identity in (54) for $\overline{R}(f)$; the identity in (55) for $\overline{R}_{1}(f)$ follows by the same argument. Define

so that $\overline{R}(f)=\lim_{\varepsilon\rightarrow 0}R_{\varepsilon}(f)$. Also, let $L(f)$ denote the right-hand side of (54).

which shows $L(f)\leq\overline{R}(f)$. Finally, it suffices to show $\{\alpha_{n}\}$ has a tight subsequence, since we can reproduce the steps above with respect to the subsequence. Towards this end, define $q_{n}({\bm{x}})=\int|{\bm{w}}^{\top}{\bm{x}}-b|d|\alpha_{n}|({\bm{w}},b)$, which is well-defined since $\alpha_{n}$ is discrete and has compact support. Then $q_{n}$ is Lipschitz with $\|q_{n}\|_{L}\leq\|\alpha_{n}\|_{1}\leq B$ for some finite $B$, hence the sequence $\{q_{n}\}$ is uniformly Lipschitz. By the Arzela-Ascoli Theorem, $\{q_{n}\}$ has a subsequence $\{q_{n_{k}}\}$ that converges uniformly on compact subsets. In particular, $q_{n_{k}}(\bm{0})=\int|b|d|\alpha_{n_{k}}|({\bm{w}},b)\leq L<\infty$ for some $L$, which implies the sequence $\{\alpha_{n_{k}}\}$ is tight.

Taking the limit as $\varepsilon\rightarrow 0$, we get $\overline{R}(f)\leq L(f)$. Therefore, we have shown $\overline{R}(f)$ is finite if and only if $L(f)$ is finite, in which case $\overline{R}(f)=L(f)$, giving the claim. ∎

The following lemma shows every infinite-width net is the pointwise limit of a sequence of finite-width nets defined in terms of sequence of measures uniformly bounded in total variation norm.

We prove the $\overline{R}(f)$ case; the $\overline{R}_{1}(f)$ case follows by the same argument. Throughout the proof we use the equivalence of $\overline{R}(f)$ given in Lemma 4, and let $\mathcal{M}(f)$ denote the right-hand side of (59).

Now we show that if $f$ is an infinite-width net, $\overline{R}_{1}(f)$ is equal to the minimal total variation norm of all even measures defining $f$ (in fact, later we show for every infinite-width net is defined in terms of a unique even measure, whose total variation norm is equal to $\overline{R}_{1}(f)$; see Lemma 10).

I Extension of ℛℛ\mathcal{R}-norm to Lipschitz functions and Proof of Theorem 1

We will need a finer characterization of the image of Schwartz functions under the dual Radon transform than what is given in Lemma 9, which is also due to Solmon (1987):

Using the above result we show the functional $\left\|f\right\|_{\mathcal{R}}$ given in Definition 1 is well-defined:

where in (64) we applied Fubini’s theorem to exchange the order of integration, whose application is justified since

and by assumption $\Delta\varphi({\bm{x}})=O(\|{\bm{x}}\|^{-d-2})$, hence $h_{+}({\bm{x}})|\Delta\varphi({\bm{x}})|=O(\|{\bm{x}}\|)^{-d-1}$, and so $\int h_{+}({\bm{x}})|\Delta\varphi({\bm{x}})|\,d{\bm{x}}<\infty$.

The following lemma shows $\left\|f\right\|_{\mathcal{R}}$ is finite if and only if $f$ is an infinite-width net, in which case $\left\|f\right\|_{\mathcal{R}}$ is given by the total variation norm of the unique even measure defining $f$.

Note that Lemma 1 is essentially a corollary of the uniqueness in the preceding result; we give the proof here for completeness.

Now we give the proof of our main theorem, which shows $\left\|f\right\|_{\mathcal{R}}=\overline{R}_{1}(f)$.

J Proof of Theorem 2

A simple calculation shows the weak gradient of $f=h_{\alpha,c}$ is given by

where $H$ is defined as $H(t)=1$ if $t\geq 0$ and $H(t)=0$ if $t<0$ otherwise. Therefore, we have

If $f({\bm{x}})={\bm{v}}_{0}^{\top}{\bm{x}}+c$ then $\overline{R}(f)=2\|{\bm{v}}_{0}\|$.

Note that $f=h_{\alpha,c}$ only if $\alpha$ is odd and $\mathcal{V}(\alpha)={\bm{v}}_{0}$. Hence, we have

we have $\overline{R}(f)=2\|{\bm{v}}_{0}\|$ as claimed. ∎

Since $\|\alpha^{+}+\alpha^{-}\|_{1}\leq\|\alpha^{+}\|_{1}+\|\alpha^{-}\|_{1}$, by Lemma 12 we see that $\overline{R}(f)\leq\|\alpha^{+}\|_{1}+2\|{\bm{v}}_{0}\|$. Now we show the lower bound. The above optimization problem is equivalent to

Finally, we show there are examples where the upper bound in Theorem 2 is attained.

Hence, $f({\bm{x}})=|{\bm{w}}_{+}^{\top}{\bm{x}}|+{\bm{w}}_{-}^{\top}{\bm{x}}$ (e.g., in 2-D one such function is $f(x,y)=x+|y|$). The dual problem for $\overline{R}(f)$ in this instance is given by:

Set ${\bm{y}}^{*}=2{\bm{w}}_{-}^{+}$, and let $\varphi^{*}$ be a continuous approximation to $\operatorname{sign}(\alpha^{+})$ whose support is localized to an arbitrarily small neighborhood of $\pm({\bm{w}}_{+},0)$. Then the pair $(\varphi^{*},{\bm{y}}^{*})$ is dual feasible since

and so $|\psi({\bm{w}},b)|\leq 1$. For these choices of $(\beta^{*},{\bm{y}}^{*})$ the dual objective is $2\|{\bm{w}}_{-}\|+\|f\|_{\mathcal{R}}$, which gives a lower bound on $\overline{R}(f)$. But this is also an upper bound on $\overline{R}(f)$ hence $\overline{R}(f)=\left\|f\right\|_{\mathcal{R}}+2\|{\bm{w}}_{-}\|$. Since $\nabla f(\infty)={\bm{w}}_{-}$, the result follows. ∎

K Properties of the ℛℛ\mathcal{R}-norm

Here we prove the properties of $\mathcal{R}$-norm discuseed in Section 4.1, including Proposition 2.

The $\mathcal{R}$-norm has the following properties:

(Scaling with dilations/contractions) Suppose $\left\|f\right\|_{\mathcal{R}}<\infty$. Let $f_{\varepsilon}({\bm{x}}):=f({\bm{x}}/\varepsilon)$, then $\|f_{\varepsilon}\|_{\mathcal{R}}=\varepsilon^{-1}\|f\|_{\mathcal{R}}$.

The 1-homogenity and triangle inequality properties follow immediate from the linearity of all operations and the definition by way of a set supremum.

To show translation invariance, define $f_{({\bm{y}})}({\bm{x}}):=f({\bm{x}}-{\bm{y}})$. Then since $\Delta$ commutes with translations we have $\Delta^{(d+1)/2}f_{({\bm{y}})}=[\Delta^{(d+1)/2}f]_{({\bm{y}})}$. Also, for any function $g$ we see that

To show rotation invariance, let $f_{{\bm{U}}}({\bm{x}})=f({\bm{U}}{\bm{x}})$ where ${\bm{U}}$ is any orthogonal $d\times d$ matrix. Then, using the fact that the Laplacian commutes with rotations, we have $\Delta^{(d+1)/2}f_{{\bm{U}}}({\bm{x}})=\Delta^{(d+1)/2}f({\bm{U}}{\bm{x}})$, and since $\mathcal{R}\{g_{{\bm{U}}}\}({\bm{w}},b)=\mathcal{R}\{g\}({\bm{U}}{\bm{w}},b)$, we see that $\mathcal{R}\{\Delta^{(d+1)/2}f_{{\bm{U}}}\}({\bm{w}},b)=\mathcal{R}\{\Delta^{(d+1)/2}f\}({\bm{U}}{\bm{w}},b)$, and so

To show the scaling under contractions/dilations (i.e., Proposition 2), let $f_{\varepsilon}({\bm{x}})=f({\bm{x}}/\varepsilon)$ for $\varepsilon>0$. Then

L Upper and Lower bounds

Here we prove several upper and lower bounds for the $\mathcal{R}$-norm. Proposition 3 is an immediate corollary of the following upper bound:

If $(-\Delta)^{(d+1)/2}f$ is a finite measure, then

In particular, if $(-\Delta)^{(d+1)/2}f$ exists in a weak sense then $\|\cdot\|_{1}$ can be interpreted as the $L^{1}$-norm.

The following result also gives a useful lower bound on the $\mathcal{R}$-norm.

Further simplifying the lower bound above gives the following.

In particular, if $\Delta f$ exists in a weak sense then $\|f\|_{\mathcal{R}}\geq\|\Delta f\|_{\infty}$.

M Radial Bump Functions

By the change of variables $t^{2}=b^{2}+r^{2}$, $t>0$, we have

where we used the fact that $\gamma_{d}c_{d}c_{d-1}=\frac{1}{(d-2)!}$.

for any $k>0$. Then a straightforward calculation using (138) gives

where $C_{d,k}=\frac{\Gamma((d-3)/2)\cdot\Gamma(1+k)}{2\Gamma((d+1)/2)+k)}$. Hence, we have $\|f\|_{\mathcal{R}}$ finite if and only if $\partial_{b}^{d}\rho(b)$ has bounded variation, which is true if and only if $k-d+\frac{d-1}{2}\geq 0$, or equivalently, $k\geq\frac{d+1}{2}$. For example, if $d=3$ then we need $k\geq 2$ in order for $\left\|f\right\|_{\mathcal{R}}$ to be finite, consistent with the previous example.

To illustrate scaling of $\left\|f\right\|_{\mathcal{R}}$ with dimension $d$, we set $k=(d+1)/2+2=(d+5)/2$ so that $\rho(b)=C_{d,(d+5)/2}(1-b^{2})^{d+2}$ for $|b|\leq 1$ and $\rho(b)=0$ otherwise. Then we can show that $|\partial^{d+1}\rho(b)|\leq|\partial^{d+1}\rho(0)|$ for $|b|\leq 1$ and $\partial^{d+1}\rho(b)=0$ for all $|b|\geq 1$. Therefore,

Performing a binomial expansion of $\rho(b)$ and taking derivatives, we obtain

for all odd $d\geq 3$. By the lower bound in Proposition 15, we also have $\left\|f\right\|_{\mathcal{R}}\geq\|\Delta f\|_{\infty}=|\Delta f(\bm{0})|=d(d+5)$. Hence $\left\|f\right\|_{\mathcal{R}}\sim d^{2}$.

N Piecewise Linear Functions

Assume $f$ is a continuous piecewise linear function with compact support satisfying assumption (a) or (b). Let $B_{1},...,B_{n}$ denote the boundaries between the regions. Since $f$ is piecewise linear and continuous, the distributional Laplacian $\Delta f$ decomposes into a linear combination of Dirac measures supported on the $d-1$ dimensional boundary sets $B_{k}$, i.e., for all smooth test functions $\varphi$ we have

We show that $\widehat{\Delta f}(\bm{\xi})$ violates the necessary decay requirements of Proposition 12 in order for $f$ to have finite $\mathcal{R}$-norm. In particular, we show under both conditions (a) and (b) there exists a ${\bm{w}}$ such that $\widehat{\Delta f}(\sigma\cdot{\bm{w}})$ is asymptotically constant as $|\sigma|\rightarrow\infty$, which gives the claim.

We first prove the claim under condition (a). Suppose, without loss of generality, that the boundary normal ${\bm{w}}_{1}$ is not parallel with all the others, i.e., ${\bm{w}}_{1}\neq{\bm{w}}_{k}$ for all $k=2,...,n$. We will write

where $F_{1}(\sigma)=c_{1}\int_{B_{1}}e^{-i2\pi\sigma{\bm{w}}_{1}^{\top}{\bm{x}}}ds({\bm{x}})$ and $F_{2}(\sigma)=\sum_{k=2}^{n}c_{k}\int_{B_{k}}e^{-i2\pi\sigma{\bm{w}}_{1}^{\top}{\bm{x}}}ds({\bm{x}})$, and give decay estimates for $F_{1}$ and $F_{2}$ separately.

First, consider $F_{1}(\sigma)$. Since ${\bm{w}}_{1}^{\top}{\bm{x}}=0$ for all ${\bm{x}}\in B_{1}$ we have

which holds for any $k=2,...,n$. Therefore, $F_{2}(\sigma)=\sum_{k=2}^{n}c_{k}\int_{B_{k}}e^{-i2\pi\sigma{\bm{w}}_{i}^{\top}{\bm{x}}}\,ds({\bm{x}})=O(1/\sigma)$ as $|\sigma|\rightarrow\infty$. This shows that $\widehat{\Delta f}(\sigma\cdot{\bm{w}}_{1})\rightarrow c_{1}s(B_{1})$, i.e., $\widehat{\Delta f}(\sigma\cdot{\bm{w}}_{1})$ is asymptotically constant, which proves the claim.

Now we prove the claim under condition $(b)$. Without loss of generality, let ${\bm{w}}_{1}$ be an inner boundary normal that is not parallel with any outer boundary normal, and assume $f$ is concave when restricted to its support. Let $I_{1}$ be the indices of all inner boundary normals parallel with ${\bm{w}}_{1}$ (including itself), let $I_{2}$ be the indices of all inner boundary normals that are not parallel with ${\bm{w}}_{1}$, and let $O$ be the indices of all outer boundary normals. Then we write