Geometric structure of shallow neural networks and constructive
  ${\mathcal L}^2$ cost minimization
In this paper, we approach the problem of cost (loss) minimization in
underparametrized shallow neural networks through the explicit construction of
upper bounds, without any use of gradient descent. A key focus is on
elucidating the geometric structure of approximate and precise minimizers. We
consider shallow neural networks with one hidden layer, a ReLU activation
function, an ${\mathcal L}^2$ Schatten class (or Hilbert-Schmidt) cost
function, input space ${\mathbb R}^M$, output space ${\mathbb R}^Q$ with $Q\leq
M$, and training input sample size $N>QM$ that can be arbitrarily large. We
prove an upper bound on the minimum of the cost function of order $O(\delta_P)$
where $\delta_P$ measures the signal to noise ratio of training inputs. In the
special case $M=Q$, we explicitly determine an exact degenerate local minimum
of the cost function, and show that the sharp value differs from the upper
bound obtained for $Q\leq M$ by a relative error $O(\delta_P^2)$. The proof of
the upper bound yields a constructively trained network; we show that it
metrizes a particular $Q$-dimensional subspace in the input space ${\mathbb
R}^M$. We comment on the characterization of the global minimum of the cost
function in the given context.
Thomas Chen, Patricia Muñoz Ewald
Geometric structure of shallow neural networks and constructive \(\mathcal{L}^{2}\) cost minimization

###### Abstract.

In this paper, we provide a geometric interpretation of the structure of shallow neural networks, characterized by one hidden layer, a ramp activation function, an \(\mathcal{L}^{2}\) Schatten class (or Hilbert-Schmidt) cost function, input space \(\mathbb{R}^{M}\), output space \(\mathbb{R}^{Q}\) with \(Q\leq M\), and training input sample size \(N>QM\). We prove an upper bound on the minimum of the cost function of order \(O(\delta_{P})\) where \(\delta_{P}\) measures the signal to noise ratio of training inputs. We obtain an approximate optimizer using projections adapted to the averages \(\overline{x_{0,j}}\) of training input vectors belonging to the same output vector \(y_{j},j=1,\ldots,Q\). In the special case \(M=Q\), we explicitly determine an exact degenerate local minimum of the cost function; the sharp value differs from the upper bound obtained for \(Q\leq M\) by a relative error \(O(\delta_{P}^{2})\). The proof of the upper bound yields a constructively trained network; we show that it metrizes the \(Q\)-dimensional subspace in the input space \(\mathbb{R}^{M}\) spanned by \(\overline{x_{0,j}}\), \(j=1,\ldots,Q\). We comment on the characterization of the global minimum of the cost function in the given context.

## 1. Introduction

The analysis of neural networks, especially Deep Learning (DL) networks, occupies a central role across a remarkably broad range of research disciplines, and the technological impact of DL algorithms in recent years has been rather monumental. However, despite of its successes, the fundamental conceptual reasons underlying their functioning are at present still insufficiently understood, and remain the subject of intense investigation, [2, 5].

The current paper is the first in a series of works in which we investigate the geometric structure of DL networks. Here, as a starting point, we analyze shallow neural networks, exhibiting one single hidden layer, and address the minimization of the cost function directly, without invoking the gradient descent flow. First, we derive an upper bound on the minimum of the cost function. As a by-product of our proof, we obtain a constructively trained shallow network which allows for a transparent geometric interpretation. In the special case of the dimensions of the input and output spaces being equal, we construct an exact degenerate local minimum of the cost function, and show that it differs from the upper bound by a subleading term. For some thematically related background, see for instance [1, 6, 5, 7, 8, 9] and the references therein. We will address an extension of our analysis to multilayer DL networks in a separate work, [3], which will invoke various results and arguments developed here.

In this paper, we will focus on the geometric structure of the system in discussion using standard mathematical terminology, and will make minimal use of specialized nomenclatures specific to the neural networks literature.
In this introductory section, we summarize the main results of this paper. We consider a shallow network for which \(y_{j}\in\mathbb{R}^{Q}\) denotes the \(j\)-th output vector, and define the output matrix

\[Y:=[y_{1},\dots,y_{Q}]\in\mathbb{R}^{Q\times Q}\,, \tag{1.1}\]

which we assume to be invertible. We assume the input space to be given by \(\mathbb{R}^{M}\) with \(M\geq Q\), and for \(j=1,\dots,Q\), we let

\[X_{0,j}:=[x_{0,j,1}\cdots x_{0,j,i}\cdots x_{0,j,N_{j}}]\in\mathbb{R}^{M\times N _{j}}\,. \tag{1.2}\]

denote the matrix of training input vectors \(x_{0,j,1}\), \(i=1,\dots,N_{j}\), which belong to the output \(y_{j}\). Letting \(N:=\sum_{j=1}^{Q}N_{j}\), the full matrix of training inputs is given by

\[X_{0}:=[X_{0,1}\cdots X_{0,j}\cdots X_{0,Q}]\in\mathbb{R}^{M\times N}\,. \tag{1.3}\]

Typically, \(Q\leq M\leq MQ<N\). We define the average of all training input vectors belonging to the output \(y_{j}\),

\[\overline{x_{0,j}}:=\frac{1}{N_{j}}\sum_{i=1}^{N_{j}}x_{0,j,i}\in\mathbb{R}^{M} \tag{1.4}\]

for \(j=1,\dots,Q\), and

\[\Delta x_{0,j,i}:=x_{0,j,i}-\overline{x_{0,j}}\,. \tag{1.5}\]

Moreover, we let

\[\delta:=\sup_{i,j}|\Delta x_{0,j,i}|\,. \tag{1.6}\]

We define

\[\Delta X_{0,j}:=[\Delta x_{0,j,1}\cdots\Delta x_{0,j,i}\cdots\Delta x_{0,j,N_{ j}}]\in\mathbb{R}^{M\times N_{j}}\,, \tag{1.7}\]

and

\[\Delta X_{0}:=[\Delta X_{0,1}\cdots\Delta X_{0,j}\cdots\Delta X_{0,Q}]\in \mathbb{R}^{M\times N}\,. \tag{1.8}\]

Moreover, we define

\[\overline{X_{0}^{red}}:=[\overline{x_{0,1}}\cdots\overline{x_{0,Q}}]\in \mathbb{R}^{M\times Q} \tag{1.9}\]

where the superscript indicates that this is the column-wise reduction of the matrix \(\overline{X_{0}}\) in (3.4), below.

We may then define the orthoprojector \(P=P^{2}=P^{T}\) onto the span of the family \(\{\overline{x_{0,j}}\}_{j=1}^{Q}\) which we assume to be linearly independent. We let \(P^{\perp}:=\mathbf{1}-P\).

We define a shallow network with hidden layer

\[x\mapsto\sigma(W_{1}x+b_{1}) \tag{1.10}\]

where \(W_{1}\in\mathbb{R}^{M\times M}\) is a weight matrix, and \(b_{1}\in\mathbb{R}^{M}\) is a shift (or bias). The activation function \(\sigma\) is assumed to act component-wise as a ramp function \((a)_{+}:=\max\{0,a\}\). We note that the requirement of \(\sigma\) acting component-wise singles out the given coordinate system in \(\mathbb{R}^{M}\) as distinct.

We define the output layer with the map \(\mathbb{R}^{M}\to\mathbb{R}^{Q}\) given by

\[x^{\prime}\mapsto W_{2}x^{\prime}+b_{2} \tag{1.11}\]

where \(W_{2}\in\mathbb{R}^{Q\times M}\) and \(b_{2}\in\mathbb{R}^{Q}\). Here, we do not include an activation function.
Then, the \(\mathcal{L}^{2}\) cost function \(\mathcal{C}[W_{j},b_{j}]\) is defined as

\[\mathcal{C}[W_{j},b_{j}]:=\sqrt{\frac{1}{N}\sum_{j=1}^{Q}\sum_{i=1}^{N_{j}}|W_{2 }\sigma(W_{1}x_{0,j,i}+b_{1})+b_{2}-y_{j}|_{\mathbb{R}^{Q}}^{2}} \tag{1.12}\]

where \(|\cdot|_{\mathbb{R}^{Q}}\) denotes the Euclidean norm on \(\mathbb{R}^{Q}\).

In Theorem 3.1, we prove an upper bound on the minimum of the cost function with the following construction.

Let \(R\in O(M)\) denote an orthogonal matrix that diagonalizes \(P\). That is, the orthoprojectors

\[P_{R}:=RPR^{T}\ \ \ \text{and}\ P_{R}^{\perp}:=RP^{\perp}R^{T} \tag{1.13}\]

are diagonal in the given coordinate system; this is important for compatibility with the fact that \(\sigma\) acts component-wise. Namely, \(R\) rotates the input data in such a way that \(\text{range}(P)\) is made to align with the coordinate axes; this decouples the action of \(\sigma\) on the rotated \(\text{range}(P)\) from its action on the rotated \(\text{range}(P^{\perp})\). That is, \(\sigma(v)=\sigma(P_{R}v)+\sigma(P_{R}^{\perp}v)\) for all \(v\in\mathbb{R}^{M}\).

Letting

\[u_{M}:=(1,1,\cdots,1)^{T}\in\mathbb{R}^{M} \tag{1.14}\]

we may choose \(\beta_{1}\geq 0\) sufficiently large (\(\beta_{1}\geq 2\max_{j,i}|x_{0,j,i}|\) is sufficient), so that the projected, translated, and rotated training input vectors \(RPx_{0,j,i}+\beta_{1}u_{M}\) are component-wise non-negative for all \(i=1,\ldots,N_{j}\), and \(j=1,\ldots,Q\).

Then, we construct an upper bound on the cost function by use of the following weights and shifts.

We choose

\[W_{1}^{*}=R \tag{1.15}\]

and \(b_{1}^{*}=P_{R}b_{1}^{*}+P_{R}^{\perp}b_{1}^{*}\) where

\[P_{R}b_{1}^{*}=\beta_{1}P_{R}u_{Q}\ \ \,\ \ P_{R}^{\perp}b_{1}^{*}=-\delta P_{R}^{ \perp}u_{M}\,, \tag{1.16}\]

with \(\beta_{1}\geq 0\) large enough to ensure component-wise non-negativity of \(RPX_{0}+P_{R}B_{1}^{*}\), where \(B_{1}^{*}=b_{1}^{*}u_{N}^{T}\), so that

\[\sigma(RPX_{0}+P_{R}B_{1}^{*})=RPX_{0}+P_{R}B_{1}^{*}\,. \tag{1.17}\]

On the other hand, the \(M-Q\) rows of \(RP^{\perp}X_{0}+P_{R}^{\perp}B_{1}^{*}\) are eliminated by way of

\[\sigma(RP^{\perp}X_{0}+P_{R}^{\perp}B_{1}^{*})=0\,. \tag{1.18}\]

We thus obtain a reduction of the dimension of the input space from \(M\) to \(Q\).

Passing to the output layer, we require \(W_{2}^{*}\) to solve

\[W_{2}^{*}R\overline{X_{0}^{red}}=Y\,, \tag{1.19}\]

which yields

\[W_{2}^{*}=Y\ \text{Pen}[\overline{X_{0}^{red}}]PR^{T}\,. \tag{1.20}\]

Here,

\[\text{Pen}[\overline{X_{0}^{red}}]:=((\overline{X_{0}^{red}})^{T}\overline{X _{0}^{red}})^{-1}(\overline{X_{0}^{red}})^{T} \tag{1.21}\]

denotes the Penrose inverse of \(\overline{X_{0}^{red}}\), which satisfies

\[\text{Pen}[\overline{X_{0}^{red}}]\overline{X_{0}^{red}}=\mathbf{1}_{Q\times Q }\ \,\ \ \overline{X_{0}^{red}}\text{Pen}[\overline{X_{0}^{red}}]=P\,. \tag{1.22}\]
Finally, we find that

\[b_{2}^{*}=-W_{2}^{*}P_{R}b_{1}^{*}\,, \tag{1.23}\]

that is, it reverts the translation by \(P_{R}b_{1}^{*}\) in the previous layer.

A pictorial way of thinking about this construction is that \(W_{1}^{*}=R\) orients the training input data with respect to the given coordinate system, in order to align it with the component-wise action of \(\sigma\). This allows for a maximal rank reduction via \(\sigma\), whereby the maximal possible amount of insignificant information is eliminated: \(P_{R}b_{1}^{*}\) pulls the significant information (in the range of \(P_{R}\)) out of the kernel of \(\sigma\), while \(P_{R}^{\perp}b_{1}^{*}\) pushes the insignificant information (in the range of \(P_{R}^{\perp}\)) into the kernel of \(\sigma\) whereby the latter is eliminated. Subsequently, \(b_{2}^{*}\) places the significant information back into its original position, and \(W_{2}^{*}\) matches it to the output matrix \(Y\) in the sense of least squares, with respect to the \(\mathcal{L}^{2}\)-norm.

We refer to the shallow network defined with these specific weights and shifts \(W_{i}^{*},b_{i}^{*}\), \(i=1,2\), as the _constructively trained network_. The parameter

\[\delta_{P}:=\sup_{j,i}\left|\mathrm{Pen}[\overline{X_{0}^{red}}]P\Delta x_{0, j,i}\right|\,, \tag{1.24}\]

measures the relative size between \(\Delta X_{0}\) and \(\overline{x_{0,j}}\), \(j=1,\ldots,Q\); in particular, \(\left|\mathrm{Pen}[\overline{X_{0}^{red}}]P\Delta x_{0,j,i}\right|\) scales like \(\frac{|\Delta x|}{|x|}\), and is scaling invariant under \(X_{0}\to\lambda X_{0}\). We then obtain

\[\min_{W_{j},b_{j}}\mathcal{C}[W_{j},b_{j}] \leq \mathcal{C}[W_{i}^{*},b_{i}^{*}] \tag{1.25}\] \[\leq \frac{1}{\sqrt{N}}\|Y\mathrm{Pen}[\overline{X_{0}^{red}}]P\Delta X _{0}\|_{\mathcal{L}^{2}}\] \[\leq \|Y\|_{op}\;\delta_{P}\;\;.\]

In order to match an arbitrary non-training input \(x\in\mathbb{R}^{M}\) with one of the output vectors \(y_{j}\), \(j\in\{1,\ldots,Q\}\), we let for \(x\in\mathbb{R}^{M}\),

\[\mathcal{C}_{j}[x]:=|W_{2}^{*}\sigma(W_{1}^{*}x+b_{1}^{*})+b_{2}^{*}-y_{j}| \tag{1.26}\]

with the weights and shifts of the constructively trained network. Then,

\[j^{*}=\mathrm{argmin}_{j}\mathcal{C}_{j}[x] \tag{1.27}\]

implies that \(x\) matches the \(j^{*}\)-th output \(y_{j^{*}}\).

We define the metric

\[d_{\widetilde{W}_{2}}(x,y):=|\widetilde{W}_{2}P(x-y)|_{\mathbb{R}^{Q}}\quad \text{for }x,y\in\mathrm{range}(P)\subset\mathbb{R}^{M} \tag{1.28}\]

where \(\widetilde{W}_{2}:=W_{2}^{*}R\), on the \(Q\)-dimensional linear subspace \(\mathrm{range}(P)\subset\mathbb{R}^{M}\). In Theorem 3.3, we prove that

\[\mathcal{C}_{j}[x]=d_{\widetilde{W}_{2}}(Px,x_{0,j})\,. \tag{1.29}\]

Therefore, matching an input \(x\in\mathbb{R}^{M}\) with an output \(y_{j^{*}}\) via the constructively trained network is equivalent to solving the metric minimization problem

\[j^{*}=\mathrm{argmin}_{j\in\{1,\ldots,Q\}}(d_{\widetilde{W}_{2}}(Px,\overline{ x_{0,j}})) \tag{1.30}\]

on the range of \(P\).

In other words, \(x\in\mathbb{R}^{M}\) is matched with the output \(y_{j^{*}}\in\mathbb{R}^{Q}\) by determining that \(Px\) is closest to \(\overline{x_{0,j^{*}}}\) among all \(\{\overline{x_{0,j}}\in\mathrm{range}(P)\subset\mathbb{R}^{M}\mid j=1,\ldots,Q\}\) in the \(d_{\widetilde{W}_{2}}\) metric.
In Theorem 3.2, we focus on the special case \(M=Q\), and present the explicit construction of a degenerate local minimum of the cost function (more precisely, a weighted version of the cost function), and obtain an improvement on the upper bound 1.25 by a factor \(1-C_{0}\delta_{P}^{2}\) for some constant \(C_{0}\geq 0\). Here, degeneracy means that for all weights and shifts \(W_{1},b_{1}\) satisfying (1.17), the corresponding minimum of the cost function attains the same value. In particular, this implies that in this range of weights and shifts, (1.25) differs from the sharp value with a relative error of order \(O(\delta_{P}^{2})\), when \(Q=M\).

In Theorem 3.5, we prove a result for \(Q=M\) closely related to Theorem 3.2, but for the case where (1.17) is not satisfied.

We expect our results for the case \(Q=M\) to be extensible to \(Q<M\), but leave a detailed analysis for future work.

This paper is organized as follows. In Section 2, we give a detailed introduction of the mathematical model describings the shallow network. In Section 3, we present the main results, namely Theorem 3.1, which we prove in Section 4, Theorem 3.3, which we prove in Section 7, and Theorem 3.5. In Section 6, we provide a detailed description of the constructively trained network.

## 2. Definition of the mathematical model

In this section, we introduce a shallow neural network, consisting of the input layer, one hidden layer, the output layer, and an \(\mathcal{L}^{2}\) cost function. Let \(Q\in\mathbb{N}\) denote the number of distinct output values. We define the output matrix

\[Y:=[y_{1},\ldots,y_{Q}]\in\mathbb{R}^{Q\times Q} \tag{2.1}\]

where \(y_{j}\in\mathbb{R}^{Q}\) is the \(j\)-th output vector. We assume that the family \(\{y_{j}\}\) is linearly independent, so that \(Y\) is invertible.

We let \(\mathbb{R}^{M}\) denote the input space with coordinate system defined by the orthonormal basis vectors \(\{e_{\ell}=(0,\ldots,0,1,0,\ldots,0)^{T}\in\mathbb{R}^{M}\,|\,\ell=1,\ldots,M\}\). For \(j\in\{1,\ldots,Q\}\), and \(N_{j}\in\mathbb{N}\),

\[x_{0,j,i}\in\mathbb{R}^{M}\ \,\ \ i\in\{1,\ldots,N_{j}\} \tag{2.2}\]

denotes the \(i\)-th training input vector corresponding to the \(j\)-th output vector \(y_{j}\). We define the matrix of all training inputs belonging to \(y_{j}\) by

\[X_{0,j}:=[x_{0,j,1}\cdots x_{0,j,i}\cdots x_{0,j,N_{j}}]\,. \tag{2.3}\]

Correspondence of a training input vector to the same output \(y_{j}\) defines an equivalence relation, that is, for each \(j\in\{1,\ldots,Q\}\) we have \(x_{0,j,i}\sim x_{0,j,i^{\prime}}\) for any \(i,i^{\prime}\in\{1,\ldots,N_{j}\}\). Accordingly, \(X_{0,j}\) labels the equivalence class of all inputs belonging to the \(j\)-th output \(y_{j}\).

Then, we define the matrix of training inputs

\[X_{0}:=[X_{0,1}\cdots X_{0,j}\cdots X_{0,Q}]\in\mathbb{R}^{M\times N} \tag{2.4}\]

where \(N:=\sum_{j=1}^{Q}N_{j}\).

We introduce weight matrices

\[W_{1}\in\mathbb{R}^{M\times M}\ \,\ \ W_{2}\in\mathbb{R}^{Q\times M} \tag{2.5}\]

and shift matrices

\[B_{1} := b_{1}u_{N}^{T}=[b_{1}\cdots b_{1}]\in\mathbb{R}^{M\times N}\] \[B_{2} := b_{2}u_{N}^{T}=[b_{2}\cdots b_{2}]\in\mathbb{R}^{Q\times N} \tag{2.6}\]
where \(b_{1}\in\mathbb{R}^{M}\) and \(b_{2}\in\mathbb{R}^{Q}\) are column vectors, and

\[u_{N}=(1,1,\ldots,1)^{T} \in\mathbb{R}^{N}\,. \tag{2.7}\]

Moreover, we choose an activation function

\[\sigma:\mathbb{R}^{M\times N} \rightarrow \mathbb{R}^{M\times N}_{+}\] \[A=[a_{ij}] \mapsto [(a_{ij})_{+}] \tag{2.8}\]

where

\[(a)_{+}:=\max\{0,a\} \tag{2.9}\]

is the standard ramp function.

We define

\[X^{(1)}:=\sigma(W_{1}X_{0}+B_{1})\in\mathbb{R}^{M\times N}_{+} \tag{2.10}\]

for the hidden layer, and

\[X^{(2)}:=W_{2}X^{(1)}+B_{2}\in\mathbb{R}^{Q\times N} \tag{2.11}\]

for the terminal layer.

We also define the extension of the matrix of outputs, \(Y^{ext}\), by

\[Y^{ext}:=[Y_{1}\cdots Y_{Q}]\in\mathbb{R}^{Q\times N} \tag{2.12}\]

where

\[Y_{j}:=[y_{j}\cdots y_{j}]\in\mathbb{R}^{Q\times N_{j}} \tag{2.13}\]

with \(N_{j}\) copies of the same output column vector \(y_{j}\). Clearly, \(Y^{ext}\) has full rank \(Q\).

Then, we consider the \(\mathcal{L}^{2}\) Schatten class (or Hilbert-Schmidt) cost function

\[\mathcal{C}[W_{j},b_{j}]:=\frac{1}{\sqrt{N}}\|X^{(2)}-Y^{ext}\|_{ \mathcal{L}^{2}} \tag{2.14}\]

with

\[\|A\|_{\mathcal{L}^{2}}\equiv\sqrt{\operatorname{Tr}(AA^{T})}\,, \tag{2.15}\]

where \(A^{T}\) is the transpose of the matrix \(A\). \(\mathcal{C}[W_{j},b_{j}]\) is a function of the weights and shifts, while the training inputs in \(X_{0}\) are considered to be a fixed set of parameters. Clearly, (2.14) is equivalent to (1.12). We note that we are not introducing an activation function in the terminal layer.

Training this shallow network amounts to finding the optimizers \(W^{**}_{j},b^{**}_{j}\) for the minimization problem

\[\mathcal{C}[W^{**}_{j},b^{**}_{j}]=\min_{W_{j},b_{j}}\mathcal{C}[W_ {j},b_{j}]\,. \tag{2.16}\]

Before we present our main results in the next section, we provide a conceptual description of our strategy.
### Discussion of the minimization process

Finding the solution to the minimization problem (2.16),

\[\mathcal{C}[W_{j}^{**},b_{j}^{**}]=\min_{W_{j},b_{j}}\frac{1}{\sqrt{N}}\|W_{2} \sigma(W_{1}X_{0}+B_{1})+B_{2}-Y^{ext}\|_{\mathcal{L}^{2}}\,, \tag{2.17}\]

one would aim at determining the minimizers \(W_{j}^{**},b_{j}^{**}\), in order to arrive at the optimally trained network.

Finding the minimum via rigorous theoretical arguments has to be contrasted with numerical minimization via the gradient descent algorithm. The gradient descent method is based on studying the dynamical system \((\mathcal{W},\Phi_{\tau})\) where

\[\mathcal{W}:=\mathbb{R}^{M\times M}\times\mathbb{R}^{Q\times M}\times\mathbb{ R}^{M}\times\mathbb{R}^{Q}\ \ \text{with}\ \ Z:=(W_{1},W_{2},b_{1},b_{2})\in\mathcal{W} \tag{2.18}\]

is the space of weights and shifts, and \(\Phi_{\tau}\) denotes the flow generated by the gradient \(G_{\mathcal{C}}[Z]:=-\nabla_{Z}\mathcal{C}[Z]\) of the cost function (or of its square), that is,

\[\partial_{\tau}Z(\tau)=G_{\mathcal{C}}[Z(\tau)]\ \,\ \ Z(\tau):=\Phi_{\tau}[Z_{0}]\ \ \,\ \ Z_{0}\in\mathcal{W}\,, \tag{2.19}\]

where \(\tau\in\mathbb{R}_{+}\). Because of

\[\partial_{\tau}\mathcal{C}[Z(\tau)]=G_{\mathcal{C}}[Z(\tau)]\cdot\partial_{t}Z (\tau)=-|G_{\mathcal{C}}[Z(\tau)]|^{2}\leq 0\,, \tag{2.20}\]

the cost function \(\mathcal{C}[Z(\tau)]\) decreases monotonically along orbits of \(\Phi_{\tau}\). Hence, the goal is to find its global minimum by numerically simulating orbits of the gradient flow. This method is highly sensitive to the choice of initial data \(Z_{0}\), which is, in practice, often picked at random, and vaguely informed by structural insights into the problem.

We make the following observations regarding the algorithmic approach.

1. If \(W_{1},b_{1}\) are given suitable initial values so that throughout the gradient descent recursion, \(W_{1}X_{0}+B_{1}\) have non-negative matrix components, then the activation function \(\sigma\), simply acts as the identity \(\mathbf{1}_{M\times M}\). Accordingly, (2.17) can be determined by solving the least squares problem \[\min_{W_{2},b_{2}}\|W_{2}X_{0}+B_{2}-Y^{ext}\|_{\mathcal{L}^{2}}\,.\] (2.21) This is because we can redefine \(W_{2}W_{1}\to W_{2}\) and \(b_{2}\to b_{2}-W_{2}b_{1}\), as \(W_{1}\) and \(b_{1}\) are unknown. The solution is given by \[W_{2}X_{0}=Y^{ext}\mathcal{P}\ \ \,\ b_{2}=0\] (2.22) where \(\mathcal{P}^{T}\) is the projector onto the range of \(X_{0}^{T}\) (which has rank \(Q\)), and \(\mathcal{P}^{\perp}=\mathbf{1}_{N\times N}-\mathcal{P}\).
2. The opposite extreme case is given when in the gradient descent algorithm, \(W_{1}\) and \(B_{1}\) are given initial values where all matrix components of \(W_{1}X_{0}+B_{1}\in\mathbb{R}^{M\times N}\) are \(\leq 0\), so that \[\sigma(W_{1}X_{0}+B_{1})=0\,.\] (2.23) Then, the cost function becomes independent of \(W_{1},b_{1}\), and its gradient in these variables is zero. One is left with the minimization problem \[\min_{b_{2}}\|B_{2}-Y^{ext}\|_{\mathcal{L}^{2}}\,,\] (2.24) which has the solution \[(b_{2})_{i}=\frac{1}{N}\sum_{j=1}^{N}[Y^{ext}]_{ij}=\sum_{j=1}^{Q}\frac{N_{j}} {N}y_{j}\,.\] (2.25)
The corresponding trained network is not capable of matching inputs \(x\in\mathbb{R}^{M}\) to outputs \(y_{j}\).
3. An intermediate variant of the previous two cases is obtained when \(W_{1}\) and \(B_{1}\) attain values where some components \((b_{1})_{i_{\ell}}\), \(\ell\in\mathcal{L}\subset\{1,\ldots,M\}\), render the corresponding rows of \([W_{1}X_{0}+B_{1}]_{i_{\ell}j}\leq 0\) for \(j=1,\ldots,N\). Then, the cost function is independent of \((b_{1})_{i_{\ell}}\) and \([W_{1}]_{i_{\ell}j}\), \(j=1,\ldots,N\), \(\ell\in\mathcal{L}\). Because by assumption, \(\operatorname{rank}(Y^{ext})=Q\), we find that if \[\operatorname{rank}(\sigma(W_{1}X_{0}+B_{1}))=Q\,,\] (2.26) the possibility of finding a solution to (2.17) remains intact, based on the surviving components of \(W_{1},b_{1}\) that occupy the non-zero rows. In this paper, we will investigate aspects of this scenario.
4. On the other hand, in the same situation as in case (3), if \[\operatorname{rank}(\sigma(W_{1}X_{0}+B_{1}))<Q\,,\] (2.27) then although one can minimize the cost function depending on the surviving variables, one will not find the global minimum (2.17).
5. Even if the initial values for \(W_{1},b_{1}\) lead to case (1), the gradient descent flow might steer them into one of the scenarios in the cases (2) \(\sim\) (4) where a rank loss is induced by \(\sigma\). Once this occurs, the gradient flow becomes independent of the components of \(W_{1},b_{1}\) with respect to which a rank loss is obtained; those components remain stationary (or plateau) at the values at which they are first eliminated by \(\sigma\).

We note that a priori, only the bound

\[\operatorname{rank}(\sigma(W_{1}X_{0}+B_{1}))\leq\operatorname{rank}((W_{1}X_ {0}+B_{1})\leq M\,, \tag{2.28}\]

is available. In applications of shallow networks, one often finds \(M>Q\) or \(M\gg Q\). The scenario described in case (3) yields the maximal reduction of complexity to still ensure compatibility with the minimization problem (2.17).

### Outline of approach

Thus, it is an interesting question to explore whether \(W_{1}\) and \(b_{1}\) can be analytically determined in a manner that a maximal rank reduction from \(M\) to \(Q\) is accomplished, as in (2.26), while maintaining the solvability of (2.17). Our analysis is based on the construction of accurate upper bounds on the cost function, and makes no recourse to gradient flow algorithms.

Our main result in Theorem 3.1, below, is an upper bound on the minimum of the cost function, obtained through the construction of a trial state as follows. Let \(P=P^{T}\) denote the orthoprojector onto the span of \(\{\overline{x_{0,j}}\in\mathbb{R}^{M}\,|\,j=1,\ldots,Q\}\), and let \(P^{\perp}\) denote its complement. We can diagonalize it with an orthogonal matrix \(R\in O(M)\)

\[P=RP_{R}R^{T}\ \,\ P^{\perp}=RP_{R}^{\perp}R^{T}\ \,\ P_{R},P_{R}^{\perp}\ \text{diagonal}, \tag{2.29}\]

where we note that \(R\) is not unique. It can be composed with any element of \(O(M)\) that leaves the ranges of \(P\) and of its complementary projector \(P^{\perp}\) invariant.

Since \(P_{R}\) is diagonal, the activation function has the additivity property

\[\sigma(v)=\sigma(P_{R}v)+\sigma(P_{\overline{R}}^{\perp}v) \tag{2.30}\]

for all \(v\in\mathbb{R}^{M}\). We split

\[b_{1}=P_{R}b_{1}+P_{\overline{R}}^{\perp}b_{1} \tag{2.31}\]
and choose

\[P_{R}b_{1}=\beta_{1}\,P_{R}u_{M}\ \ \in\mathbb{R}^{M} \tag{2.32}\]

We choose \(\beta_{1}\geq 0\) large enough that

\[\sigma(RPx_{0,j,i}+P_{R}b_{1})=RPx_{0,j,i}+P_{R}b_{1} \tag{2.33}\]

that is, \(P_{R}b_{1}\) translates the projected and rotated training inputs in the direction of \(P_{R}u_{M}\) sufficiently far from the origin, to render all components non-negative. For this purpose, \(\beta_{1}\geq 2\max_{j,i}|x_{0,j,i}|\) is sufficient.

Then, we let \(W_{1}=R\), so that

\[\sigma(W_{1}X_{0}+B_{1})=\sigma(RPX_{0}+P_{R}B_{1})+\sigma(RP^{ \perp}X_{0}+P_{R}^{\perp}B_{1})\,. \tag{2.34}\]

Here we used that \(RP=P_{R}RP\) and \(RP^{\perp}=P_{R}^{\perp}RP^{\perp}\), and (2.30). For the second term on the r.h.s., we observe that \(P^{\perp}X_{0}=P^{\perp}\Delta X_{0}\), therefore choosing

\[P_{R}^{\perp}b_{1}=-\delta P_{R}^{\perp}u_{M}\,, \tag{2.35}\]

where we recall \(\delta\) from (1.6) and \(u_{M}\) from (1.14), we find

\[\sigma(RP^{\perp}\Delta X_{0}+P_{R}^{\perp}B_{1})=0 \tag{2.36}\]

as all components in its argument are shifted to non-positive values by (2.35). This reduces the rank of (2.34) from \(M\) to \(Q\).

Therefore, (2.34) reduces to

\[\sigma(W_{1}X_{0}+B_{1})=\sigma(RPX_{0}+P_{R}B_{1})=RPX_{0}+P_{R}B_{1}\,. \tag{2.37}\]

This is because \(R\) and \(P_{R}b_{1}\) have been chosen in a manner that \(RPX_{0}+P_{R}B_{1}\) has non-negative matrix components. Therefore, minimization of the cost function reduces to determining

\[\min_{W_{2},b_{1},b_{2}}\frac{1}{\sqrt{N}}\|W_{2}RPX_{0}+W_{2}P_{ R}B_{1}+B_{2}-Y^{ext}\|_{\mathcal{L}^{2}} \tag{2.38}\]

We may redefine \(b_{2}\to b_{2}-W_{2}P_{R}b_{1}\), which further reduces the problem to finding

\[\min_{W_{2},b_{2}}\frac{1}{\sqrt{N}}\|W_{2}RPX_{0}+B_{2}-Y^{ext} \|_{\mathcal{L}^{2}}\,. \tag{2.39}\]

This is a least squares problem from which we deduce an upper bound on the minimum of the cost function in Theorem 3.1. Theorem 3.3 provides a very natural and transparent geometric characterization of the shallow network trained with these values of \(W_{i},b_{i}\); namely, it metrizes the range of \(P\) in the input space \(\mathbb{R}^{M}\), and given an arbitrary input \(x\), it measures proximity of \(Px\) to the vectors \(\overline{x_{0,j}}\), \(j=1,\dots,Q\).

In the special case \(M=Q\), we are able to explicitly determine a degenerate local minimum of the cost function, and arrive at an improvement on the upper bound obtained in Theorem 3.2, by a factor \(1-C_{0}\delta_{P}^{2}\) for some constant \(C_{0}\geq 0\). In this construction, the activation function does not truncate the training input matrix.

In Theorem 3.5, we prove a result for \(Q=M\) similar to Theorem 3.2, but for the case in which the activation function truncates the training input matrix in a manner that (2.34) does not hold.
## 3. Statement of Main Results

In this section, we present the main results of this paper, an explicit upper bound on the cost function in Theorem 3.1 which leads to the construction of an optimized constructively trained shallow network, and its natural geometric interpretation in Theorem 3.3. In Theorem 3.2, we construct an explicit degenerate local minimum of the cost function, for the special case of input and output space having equal dimension, \(M=Q\). In Theorem 3.5, we address the effects of truncation of the training input matrix due to the activation function, in the case \(Q=M\).

### Upper bound on minimum of cost function for \(M>q\)

Let

\[\overline{x_{0,j}}:=\frac{1}{N_{j}}\sum_{i=1}^{N_{j}}x_{0,j,i}\in\mathbb{R}^{M} \tag{3.1}\]

denote the average over the equivalence class of all training input vectors corresponding to the \(j\)-th output, and

\[\Delta x_{0,j,i}:=x_{0,j,i}-\overline{x_{0,j}}\,. \tag{3.2}\]

Moreover, let

\[\overline{X_{0,j}}:=[\overline{x_{0,j}}\cdots\overline{x_{0,j}}]\in\mathbb{R} ^{M\times N_{j}} \tag{3.3}\]

and

\[\overline{X_{0}}:=[\overline{X_{0,1}}\cdots\overline{X_{0,Q}}]\in\mathbb{R}^{ M\times N}\,. \tag{3.4}\]

Similarly, let

\[\Delta X_{0,j}:=[\Delta x_{0,j,1}\cdots\Delta x_{0,j,N_{j}}]\in\mathbb{R}^{M \times N_{j}}\,. \tag{3.5}\]

and

\[\Delta X_{0}:=[\Delta X_{0,1}\cdots\Delta X_{0,Q}]\in\mathbb{R}^{M\times N}\,. \tag{3.6}\]

Then,

\[X_{0}=\overline{X_{0}}+\Delta X_{0}\,, \tag{3.7}\]

and we let

\[\delta:=\sup_{j,i}|\Delta x_{0,j,i}|\,\,\,. \tag{3.8}\]

Next, we define the reduction of \(\overline{X_{0}}\) as

\[\overline{X_{0}^{red}}:=[\overline{x_{0,1}}\cdots\overline{x_{0,Q}}]\in \mathbb{R}^{M\times Q} \tag{3.9}\]

and assume that its rank is maximal, \(\operatorname{rank}(\overline{X_{0}^{red}})=Q\). The latter condition is necessary for the averages of training inputs to be able to distinguish between the different output values \(y_{j}\), \(j=1,\dots,Q\).

Because of \(\operatorname{rank}(\overline{X_{0}^{red}})=Q\), the matrix

\[(\overline{X_{0}^{red}})^{T}\overline{X_{0}^{red}}\in\mathbb{R}^{Q\times Q} \tag{3.10}\]

is invertible, and we can define the Penrose inverse of \(\overline{X_{0}^{red}}\),

\[\operatorname{Pen}[\overline{X_{0}^{red}}]:=\big{(}(\overline{X_{0}^{red}})^{ T}\overline{X_{0}^{red}}\big{)}^{-1}(\overline{X_{0}^{red}})^{T}\in\mathbb{R}^{Q \times M} \tag{3.11}\]

The orthoprojector onto the range of \(\overline{X_{0}^{red}}\) is then given by

\[P:=\overline{X_{0}^{red}}\operatorname{Pen}[\overline{X_{0}^{red}}]\in \mathbb{R}^{M\times M} \tag{3.12}\]
and we note that \(\mathrm{Pen}[\overline{X_{0}^{red}}]\overline{X_{0}^{red}}=\mathbf{1}_{Q\times Q}\). The projector property \(P^{2}=P\) is thus easily verified, and orthogonality with respect to the Euclidean inner product \((\cdot,\cdot)\) on \(\mathbb{R}^{M}\) holds due to \(P^{T}=P\), whereby \((v,Pw)=(Pv,w)\) for any \(v,w\in\mathbb{R}^{M}\). In particular, we have

\[P\overline{X_{0}^{red}}=\overline{X_{0}^{red}}\ \ \text{and}\ \ \ P\overline{X_{0}}= \overline{X_{0}}\,, \tag{3.13}\]

by construction.

Moreover, we define

\[\delta_{P}:=\sup_{i,j}\left|\mathrm{Pen}[\overline{X_{0}^{red}}]P\Delta x_{0, j,i}\right|\,, \tag{3.14}\]

which measures the relative size between \(\overline{X_{0}^{red}}\) and \(P\Delta X_{0}\), as \(\left|\mathrm{Pen}[\overline{X_{0}^{red}}]P\Delta x_{0,j,i}\right|\) scales like the noise to signal ratio \(\frac{|\Delta x|}{|x|}\) of training inputs.

**Theorem 3.1**.: _Let \(Q\leq M\leq QM<N\). Assume that \(R\in O(M)\) diagonalizes \(P,P^{\perp}\), and let \(\beta_{1}\geq 2\max_{j,i}|x_{0,j,i}|\)._

_Let \(\mathcal{C}[W_{i}^{*},b_{i}^{*}]\) be the cost function evaluated for the trained shallow network defined by the following weights and shifts,_

\[W_{1}^{*}=R\,, \tag{3.15}\]

_and_

\[W_{2}^{*}=\widetilde{W}_{2}R^{T}\ \ \text{where}\ \widetilde{W}_{2}=Y\ \mathrm{Pen}[\overline{X_{0}^{red}}]P\,. \tag{3.16}\]

_Moreover, \(b_{1}^{*}=P_{R}b_{1}^{*}+P_{R}^{\perp}b_{1}^{*}\) with_

\[P_{R}b_{1}^{*}=\beta_{1}u_{M}\ \,\ P_{R}^{\perp}b_{1}^{*}=-\delta P_{R}^{\perp}u_{M} \ \in\mathbb{R}^{M}\,, \tag{3.17}\]

_for \(u_{M}\in\mathbb{R}^{M}\) as in (1.14), and_

\[b_{2}^{*}=-W_{2}^{*}P_{R}b_{1}^{*}\ \ \in\mathbb{R}^{Q}\,. \tag{3.18}\]

_Then, the minimum of the cost function satisfies the upper bound_

\[\min_{W_{j},b_{j}}\mathcal{C}[W_{j},b_{j}]\leq\mathcal{C}[W_{i}^{*},b_{i}^{*}] \leq\frac{1}{\sqrt{N}}\|Y\ \mathrm{Pen}[\overline{X_{0}^{red}}]P\Delta X_{0}\|_{\mathcal{L}^{2}}\,, \tag{3.19}\]

_which implies_

\[\min_{W_{j},b_{j}}\mathcal{C}[W_{j},b_{j}]\ \leq\ \|Y\|_{op}\ \delta_{P} \tag{3.20}\]

_where \(N=\sum_{\ell=1}^{Q}N_{\ell}\)._

We observe that, for any \(\lambda>0\), the rescaling of training inputs

\[x_{0,j,i}\to\lambda x_{0,j,i}\ \ \forall j=1,\ldots,Q\ \,\ i=1,\ldots,N_{j}\,, \tag{3.21}\]

induces

\[\mathrm{Pen}[\overline{X_{0}^{red}}]\to\lambda^{-1}\mathrm{Pen}[\overline{X_{ 0}^{red}}]\ \ \,\ \Delta X_{0}\to\lambda\Delta X_{0}\,. \tag{3.22}\]

Therefore, the upper bound in (3.19) is scaling invariant, as it only depends on the signal to noise ratio of training input data, which is controlled by \(\delta_{P}\).
### Exact degenerate local minimum in the case \(M=q\)

We are able to explicitly determine an exact local degenerate minimum of the cost function in the case \(Q=M\) where the input and output spaces have the same dimension; here, we show that the upper bound obtained above differs from the sharp value by a relative error of order \(O(\delta_{P}^{2})\). We note that in this situation, \(P=\mathbf{1}_{Q\times Q}\) and \(P^{\perp}=0\). Moreover, \(\overline{X_{0}^{red}}\) is invertible, and therefore, \(\mathrm{Pen}[\overline{X_{0}^{red}}]=(\overline{X_{0}^{red}})^{-1}\). We prove a result adapted to a weighted variant of the cost function defined as follows. Let \(\mathcal{N}\in\mathbb{R}^{N\times N}\) be the block diagonal matrix given by

\[\mathcal{N}:=\mathrm{diag}(N_{j}\mathbf{1}_{N_{j}\times N_{j}}\,|\,j=1,\dots, Q)\,. \tag{3.23}\]

We introduce the inner product on \(\mathbb{R}^{Q\times N}\)

\[(A,B)_{\mathcal{L}_{\mathcal{N}}^{2}}:=\mathrm{Tr}(A\mathcal{N}^{-1}B^{T}) \tag{3.24}\]

and

\[\|A\|_{\mathcal{L}_{\mathcal{N}}^{2}}:=\sqrt{(A,A)_{\mathcal{L}_{\mathcal{N} }^{2}}}\,. \tag{3.25}\]

We define the weighted cost function

\[\mathcal{C}_{\mathcal{N}}[W_{i},b_{i}]:=\|X^{(2)}-Y^{ext}\|_{\mathcal{L}_{ \mathcal{N}}^{2}}\,. \tag{3.26}\]

This is equivalent to

\[\mathcal{C}_{\mathcal{N}}[W_{i},b_{i}]:=\sqrt{\sum_{j=1}^{Q}\frac{1}{N_{j}} \sum_{i=1}^{N_{j}}|W_{2}\sigma(W_{1}x_{0,j,i}+b_{1})+b_{2}-y_{j}|_{\mathbb{R}^ {Q}}^{2}}\,. \tag{3.27}\]

The weights \(\frac{1}{N_{j}}\) ensure that the contributions to the cost function belonging to inputs \(y_{j}\) do not depend on their sample sizes \(N_{j}\). We note that for uniform sample sizes, where \(\frac{N_{j}}{N}=\frac{1}{Q}\ \forall j\), we have that \(\mathcal{C}_{\mathcal{N}}[W_{i},b_{i}]=\sqrt{Q}\mathcal{C}[W_{i},b_{i}]\).

**Theorem 3.2**.: _Assume \(M=Q<MQ<N\). Then, if \(\mathrm{rank}(\overline{X_{0}})=Q\), let_

\[\mathcal{P}:=\mathcal{N}^{-1}X_{0}^{T}(X_{0}\mathcal{N}^{-1}X_{0}^{T})^{-1}X_{ 0}\ \in\mathbb{R}^{N\times N}\,. \tag{3.28}\]

_Its transpose \(\mathcal{P}^{T}\) is an orthonormal projector in \(\mathbb{R}^{N}\) with respect to the inner product \(\langle u,v\rangle_{\mathcal{N}}:=(u,\mathcal{N}^{-1}v)\). In particular, \((A\mathcal{P},B)_{\mathcal{L}_{\mathcal{N}}^{2}}=(A,B\mathcal{P})_{\mathcal{ L}_{\mathcal{N}}^{2}}\) for all \(A,B\in\mathbb{R}^{Q\times N}\). Letting \(\mathcal{P}^{\perp}=\mathbf{1}_{N\times N}-\mathcal{P}\), the weighted cost function satisfies the upper bound_

\[\min_{W_{j},b_{j}}\mathcal{C}_{\mathcal{N}}[W_{j},b_{j}] \leq \mathcal{C}_{\mathcal{N}}[W_{i}^{*},b_{i}^{*}] \tag{3.29}\] \[= \|Y^{ext}\mathcal{P}^{\perp}\|_{\mathcal{L}_{\mathcal{N}}^{2}}\] \[= \left\|Y|\Delta_{2}^{rel}|^{\frac{1}{2}}\big{(}1+\Delta_{2}^{rel} \big{)}^{-\frac{1}{2}}\right\|_{\mathcal{L}^{2}}\] \[\leq (1-C_{0}\delta_{P}^{2})\ \|Y\ (\overline{X_{0}^{red}})^{-1}\Delta X_{0}\|_{ \mathcal{L}_{\mathcal{N}}^{2}}\,,\]

_for a constant \(C_{0}\geq 0\), and where_

\[\Delta_{2}^{rel}:=\Delta_{1}^{rel}\mathcal{N}^{-1}(\Delta_{1}^{rel})^{T}\ \,\ \mathrm{with}\ \Delta_{1}^{rel}:=(\overline{X_{0}^{red}})^{-1}\Delta X_{0}\,. \tag{3.30}\]

_The weights and shifts realizing the upper bound are given by_

\[W_{1}^{*}=\mathbf{1}_{Q\times Q} \tag{3.31}\]

_and_

\[W_{2}^{*}=Y(\overline{X_{0}^{red}})^{T}(X_{0}\mathcal{N}^{-1}X_{0}^{T})^{-1} \in\mathbb{R}^{Q\times Q}\,. \tag{3.32}\]
_Moreover,_

\[b_{1}^{*}=\beta_{1}u_{Q}\ \,\ b_{2}^{*}=-W_{2}^{*}b_{1}^{*}\,, \tag{3.33}\]

_with \(u_{Q}\in\mathbb{R}^{Q}\) as in (1.14), and where \(\beta_{1}\geq 2\max_{j,i}|x_{0,j,i}|\) so that_

\[\sigma(W_{1}^{*}X_{0}+B_{1}^{*})=W_{1}^{*}X_{0}+B_{1}^{*}\,. \tag{3.34}\]

_Notably,_

\[W_{2}^{*}=\widetilde{W}_{2}+O(\delta_{P}^{2}) \tag{3.35}\]

_differs by \(O(\delta_{P}^{2})\) from \(\widetilde{W}_{2}\) in (3.16) used for the upper bound (3.19)._

_In particular, \(\mathcal{C}_{\mathcal{N}}[W_{i}^{*},b_{i}^{*}]\) is a local minimum of the cost function; it is degenerate, assuming the same value for all \(W_{i},b_{i}\) such that \(W_{1},b_{1}\) satisfy the condition (3.34)._

_Moreover, \(\mathcal{C}_{\mathcal{N}}[W_{i}^{*},b_{i}^{*}]\) is invariant under reparametrizations of the training inputs \(X_{0}\to KX_{0}\), for all \(K\in GL(Q)\)._

The proof is given in Section 5.

### Geometric interpretation

To match an arbitrary, non-training input \(x\in\mathbb{R}^{M}\) with one of the output vectors \(y_{j}\), \(j\in\{1,\ldots,Q\}\), let for \(x\in\mathbb{R}^{M}\),

\[\mathcal{C}_{j}[x]:=|W_{2}^{*}(W_{1}^{*}x+b_{1}^{*})_{+}+b_{2}^{*}-y_{j}|\,. \tag{3.36}\]

Here, \((\cdot)_{+}\) denotes the ramp function (2.9), acting component-wise on a vector.

Given an arbitrary non-training input \(x\in\mathbb{R}^{M}\),

\[j^{*}=\operatorname{argmin}_{j}\mathcal{C}_{j}[x] \tag{3.37}\]

implies that \(x\) matches the \(j^{*}\)-th output \(y_{j^{*}}\).

The proof of Theorem 3.1 provides us with a particular set of weights \(W_{1}^{*},W_{2}^{*}\) and shifts \(b_{1}^{*},b_{2}^{*}\) through an explicit construction, which yield the upper bound (3.19). We will refer to the shallow network trained with this choice of \(W_{i}^{*},b_{i}^{*}\) as the _constructively trained shallow network_. A detailed discussion is provided in Section 6.

**Theorem 3.3**.: _Assume \(Q\leq M\leq QM<N\). Let \(W_{i}^{*},b_{i}^{*}\), \(i=1,2\), denote the weights and shifts determined in Theorem 3.1. Let_

\[\widetilde{W}_{2}:=Y\operatorname{Pen}[\overline{X_{0}^{red}}]\in\mathbb{R}^ {Q\times M}\,, \tag{3.38}\]

_and define the metric_

\[d_{\widetilde{W}_{2}}(x,y):=|\widetilde{W}_{2}P(x-y)|\ \ \text{ for }x,y\in\operatorname{range}(P) \tag{3.39}\]

_on the \(Q\)-dimensional linear subspace \(\operatorname{range}(P)\subset\mathbb{R}^{M}\), where \(|\cdot|\) denotes the Euclidean norm on \(\mathbb{R}^{Q}\). Then,_

\[\mathcal{C}_{j}[x]=d_{\widetilde{W}_{2}}(Px,\overline{x_{0,j}}) \tag{3.40}\]

_and matching an input \(x\in\mathbb{R}^{M}\) with an output \(y_{j^{*}}\) via the constructively trained shallow network is equivalent to the solution of the metric minimization problem_

\[j^{*}=\operatorname{argmin}_{j\in\{1,\ldots,Q\}}(d_{\widetilde{W}_{2}}(Px, \overline{x_{0,j}})) \tag{3.41}\]

_on the range of \(P\)._
The proof is given in Section 7.

First of all, we note that \(\widetilde{W}_{2}P=\widetilde{W}_{2}\) has full rank \(Q\), therefore \(x\mapsto|\widetilde{W}_{2}Px|^{2}\) is a non-degenerate quadratic form on the range of \(P\). Therefore, (3.39) indeed defines a metric on the range of \(P\).

The constructively trained shallow network thus obtained matches a non-training input \(x\in\mathbb{R}^{M}\) with an output vector \(y_{j^{*}}\) by splitting

\[x=Px+P^{\perp}x\,, \tag{3.42}\]

and by determining which of the average training input vectors \(\overline{x_{0,j}}\), \(j=1,\ldots,Q\), is closest to \(Px\) in the \(d_{\widetilde{W}_{2}}\) metric. The trained shallow network cuts off the components \(P^{\perp}x\). This procedure is geometrically natural and transparent.

### Dependence on truncation

Our next result addresses the effect of truncations enacted by the activation function \(\sigma\) in the case \(M=Q\). For the results in Theorem 3.2, we assumed that \(W_{1},b_{1}\) are in the region (depending on \(X_{0}\)) such that

\[\sigma(W_{1}X_{0}+B_{1})=W_{1}X_{0}+B_{1} \tag{3.43}\]

holds. Here we discuss the situation in which \(\sigma\) acts nontrivially. In this case, we assume that \(W_{1},b_{1}\) are given, with \(W_{1}\in\mathbb{R}^{Q\times Q}\) invertible, and observe that all matrix components of

\[X^{(1)}=\sigma(W_{1}X_{0}+B_{1}) \tag{3.44}\]

are non-negative, as \(X^{(1)}\) lies in the image of \(\sigma\), and hence,

\[\sigma(X^{(1)})=X^{(1)}\,. \tag{3.45}\]

We define the truncation map \(\tau_{W_{1},b_{1}}\) as follows.

**Definition 3.4**.: _Let \(W_{1}\in GL(Q)\), \(b_{1}\in\mathbb{R}^{Q}\). Then, the truncation map \(\tau_{W_{1},b_{1}}:\mathbb{R}^{Q\times N}\to\mathbb{R}^{Q\times N}\) is defined by_

\[\tau_{W_{1},b_{1}}(X_{0}) := W_{1}^{-1}(\sigma(W_{1}X_{0}+B_{1})-B_{1}) \tag{3.46}\] \[= W_{1}^{-1}(X^{(1)}-B_{1})\,.\]

_That is, \(\tau_{W_{1},b_{1}}=a_{W_{1},b_{1}}^{-1}\circ\sigma\circ a_{W_{1},b_{1}}\) under the affine map \(a_{W_{1},b_{1}}:X_{0}\mapsto W_{1}X_{0}+B_{1}\)._

_We say that \(\tau_{W_{1},b_{1}}\) is rank preserving (with respect to \(X_{0}\)) if both_

\[\mathrm{rank}(\tau_{W_{1},b_{1}}(X_{0})) = \mathrm{rank}(X_{0})\] \[\mathrm{rank}(\overline{\tau_{W_{1},b_{1}}(X_{0})}) = \mathrm{rank}(\overline{X_{0}}) \tag{3.47}\]

_hold, and that it is rank reducing otherwise._

Then, we verify that

\[\sigma(W_{1}\tau_{W_{1},b_{1}}(X_{0})+B_{1}) = \sigma(X^{(1)})=X^{(1)} \tag{3.48}\] \[= W_{1}\tau_{W_{1},b_{1}}(X_{0})+B_{1}\,.\]

This means that while the condition (3.43) does not hold for \(X_{0}\), it does hold for \(\tau_{W_{1},b_{1}}(X_{0})\).

Accordingly, we can define the matrices

\[\overline{\tau_{W_{1},b_{1}}(X_{0})}\,\,\,,\,\overline{(\tau_{W_{1},b_{1}}(X_ {0}))^{red}}\,\,\,,\,\Delta(\tau_{W_{1},b_{1}}(X_{0})) \tag{3.49}\]

in analogy to \(\overline{X_{0}},\overline{X_{0}^{red}},\Delta X_{0}\).
If no rank reduction is induced by the truncation, \(\overline{(\tau_{W_{1},b_{1}}(X_{0}))^{red}}\) is invertible, and we obtain the following theorem.

**Theorem 3.5**.: _Let \(Q=M\), and assume that the truncation map \(\tau_{W_{1},b_{1}}(X_{0})\) is rank preserving. Then, for any fixed \((W_{1},b_{1})\in GL(Q)\times\mathbb{R}^{Q}\),_

\[\min_{W_{2},b_{2}}\mathcal{C}_{\mathcal{N}}[W_{i},b_{i}]=\|Y^{ext}\mathcal{P}_ {\tau_{W_{1},b_{1}}(X_{0})}^{\perp}\|_{\mathcal{L}_{\mathcal{N}}^{2}} \tag{3.50}\]

_where the projector \(\mathcal{P}_{\tau_{W_{1},b_{1}}(X_{0})}\) is obtained from (3.28) by substituting the truncated training inputs \(\tau_{W_{1},b_{1}}(X_{0})\) for \(X_{0}\). In particular, one explicitly has_

\[\min_{W_{2},b_{2}}\mathcal{C}_{\mathcal{N}}[W_{i},b_{i}] = \left\|Y|\Delta_{2}^{rel,tr}|^{\frac{1}{2}}\big{(}1+\Delta_{2}^{ rel,tr}\big{)}^{-\frac{1}{2}}\right\|_{\mathcal{L}^{2}} \tag{3.51}\] \[\leq (1-C_{0}\delta_{P,tr}^{2})\|Y\Delta_{1}^{rel,tr}\|_{\mathcal{L}_ {\mathcal{N}}^{2}}\]

_where_

\[\Delta_{2}^{rel,tr} := \Delta_{1}^{rel,tr}\mathcal{N}^{-1}(\Delta_{1}^{rel,tr})^{T}\;\;, \;\text{with}\] \[\Delta_{1}^{rel,tr} := (\overline{(\tau_{W_{1},b_{1}}(X_{0}))^{red}})^{-1}\Delta(\tau_{ W_{1},b_{1}}(x_{0,j,i}))\,, \tag{3.52}\]

_in analogy to (3.29), and for some \(C_{0}\geq 0\), where_

\[\delta_{P,tr}:=\sup_{j,i}\left|(\overline{(\tau_{W_{1},b_{1}}(X_{0}))^{red}}) ^{-1}\Delta(\tau_{W_{1},b_{1}}(x_{0,j,i}))\right| \tag{3.53}\]

_measures the signal to noise ratio of the truncated training input data._

Proof.: Due to (3.48), we may follow the proof of Theorem 3.2 verbatim, and we find that for the given fixed choice of \(W_{1},b_{1}\), minimization in \(W_{2},b_{2}\) yields (3.50) and (3.51). 

In general, (3.51) does not constitute a stationary solution with respect to \(W_{1},b_{1}\), as the right hand sides of (3.50), (3.51) generically depend nontrivially on \(W_{1},b_{1}\). It thus remains to determine the infimum of (3.50), (3.51) with respect to \(W_{1},b_{1}\), in order to find a candidate for the global minimum. The latter, however, might depend sensitively on the detailed properties of \(\Delta X_{0}\), which are random.

On the other hand, in the situation addressed in Theorem 3.2, \(W_{1},b_{1}\) are contained in the parameter region \(\mathcal{F}_{X_{0}}\) (depending on \(X_{0}\)) defined by the fixed point relation for the truncation map (equivalent to (3.43)),

\[\mathcal{F}_{X_{0}}:=\left\{(W_{1},b_{1})\in GL(Q)\times\mathbb{R}^{Q}\mid \tau_{W_{1},b_{1}}(X_{0})=X_{0}\right\}. \tag{3.54}\]

As a result, we indeed obtained a degenerate stationary solution with respect to \(W_{1},b_{1}\),

\[\min_{(W_{1},b_{1})\in\mathcal{F}_{X_{0}};W_{2},b_{2}}\mathcal{C}_{\mathcal{N }}[W_{i},b_{i}]=\|Y^{ext}\mathcal{P}^{\perp}\|_{\mathcal{L}_{\mathcal{N}}^{2}} \tag{3.55}\]

in (3.29), with \(\mathcal{P}\) given in (3.28). Hence, \(\mathcal{F}_{X_{0}}\) parametrizes an invariant manifold of equilibria of the gradient descent flow, which corresponds to a level surface of the cost function.

**Remark 3.6**.: _The global minimum of the cost function has either the form (3.50) for an optimizer \((W_{1}^{**},b_{1}^{**})\) (with a rank preserving truncation), or (3.55). This is because any other scenario employs a rank reducing truncation. Since \(Y^{ext}\) has rank \(Q\), minimization of (3.26) under the constraint \(\operatorname{rank}(\tau_{W_{1},b_{1}}(X_{0}))<Q\) cannot yield a global minimum._
_The key point in minimizing (3.50) with respect to \((W_{1},b_{1})\) is to determine a rank preserving truncation which minimizes the signal to noise ratio of the truncated training inputs. Whether for a local or global minimum of the form (3.50) or (3.55), respectively, a shallow network trained with the corresponding weights and shifts \(W_{i},b_{i}\), \(i=1,2\), will be able to match inputs \(x\in\mathbb{R}^{Q}\) to outputs \(y_{j}\), through a straightforward generalization of Theorem 3.3._

We expect these considerations to carry over to the general case \(Q<M\), but leave a detailed discussion to future work.

## 4. Proof of Theorem 3.1

Let \(R\in O(M)\) diagonalize \(P,P^{\perp}\) (which are symmetric),

\[P_{R}:=RPR^{T}\quad,\ \ P_{R}^{\perp}:=RP^{\perp}R^{T}\,, \tag{4.1}\]

where \(P_{R}\) has \(Q\) diagonal entries of \(1\), and all other components are \(0\), while \(P_{R}^{\perp}\) has \(M-Q\) diagonal entries of \(1\), and all other components are \(0\). This does not characterize \(R\) uniquely, as we may compose \(R\) with an arbitrary \(R^{\prime}\in O(M)\) that leaves the ranges of \(P,P^{\perp}\) invariant. Diagonality of \(P_{R},P_{R}^{\perp}\) implies that the activation function, which acts component-wise as the ramp function (2.9), satisfies

\[\sigma(P_{R}v+P_{R}^{\perp}v)=\sigma(P_{R}v)+\sigma(P_{R}^{\perp}v) \tag{4.2}\]

for any \(v\in\mathbb{R}^{M}\), and where of course, \(P_{R}+P_{R}^{\perp}=\mathbf{1}_{M\times M}\).

We write

\[b_{1}=P_{R}b_{1}+P_{R}^{\perp}b_{1} \tag{4.3}\]

where we choose

\[P_{R}b_{1}=\beta_{1}u_{Q}\,, \tag{4.4}\]

and we pick

\[\beta_{1}\geq 2\rho\ \,\ \ \rho:=\max_{j,i}|x_{0,j,i}|\,. \tag{4.5}\]

This ensures that \(RPx_{0,j,i}+P_{R}b_{1}\in B_{\rho}(\beta_{1}u_{Q})\subset P_{R}\mathbb{R}_{+}^ {M}\) so that

\[\sigma(RPx_{0,j,i}+P_{R}b_{1})=RPx_{0,j,i}+P_{R}b_{1}\quad,\ \ j=1,\ldots,Q\ \,,\ \forall\ i=1,\ldots,N_{j} \tag{4.6}\]

To construct the upper bound, we choose

\[W_{1}=R\,. \tag{4.7}\]

Then,

\[X^{(1)} = \sigma(W_{1}X_{0}+B_{1}) \tag{4.8}\] \[= \sigma((P_{R}+P_{R}^{\perp})RX_{0}+(P_{R}+P_{R}^{\perp})B_{1})\] \[= \sigma(P_{R}RX_{0}+P_{R}B_{1})+\sigma(P_{R}^{\perp}RX_{0}+P_{R}^ {\perp}B_{1})\] \[= \sigma(RPX_{0}+P_{R}B_{1})+\sigma(RP^{\perp}\Delta X_{0}+P_{R}^{ \perp}B_{1})\,.\]

This is because by construction,

\[P\overline{X_{0}}=\overline{X_{0}}\ \,\ \ P^{\perp}\overline{X_{0}}=0\,. \tag{4.9}\]

Moreover, we used

\[P_{R}R=RP\ \,\ \ P_{R}^{\perp}R=RP^{\perp}\,, \tag{4.10}\]

from (4.1).
Going to the next layer, we have

\[X^{(2)}=W_{2}X^{(1)}+B_{2}\,. \tag{4.11}\]

Our goal is to minimize the expression on the right hand side of

\[\sqrt{N}{\mathcal{C}}[W_{j},b_{j}] \leq \|W_{2}\sigma(RPX_{0}+P_{R}B_{1})+B_{2}-Y^{ext}\|_{{\mathcal{L}}^{ 2}} \tag{4.12}\] \[\qquad\qquad+\|W_{2}\sigma(RP^{\perp}\Delta X_{0}+P_{R}^{\perp}B_ {1})\|_{{\mathcal{L}}^{2}}\,.\]

We note that because \({\rm rank}(Y^{ext})=Q\), and

\[{\rm rank}(\sigma(\underbrace{RPX_{0}}_{=P_{R}RPX_{0}}+P_{R}B_{1}))\leq Q\,, \tag{4.13}\]

minimization of the first term on the r.h.s. of (4.12) requires that the latter has maximal rank \(Q\). Due to our choice of \(R\) and \(P_{R}b_{1}\) which imply that (4.6) holds, we find that

\[\sigma(RPX_{0}+P_{R}B_{1}) = RPX_{0}+P_{R}B_{1} \tag{4.14}\] \[= P_{R}R(\overline{X_{0}}+P\Delta X_{0})+P_{R}B_{1}\,.\]

To minimize the first term on the r.h.s. of (4.12), one might want to try to solve

\[W_{2}S=Y^{ext} \tag{4.15}\]

where

\[S:=P_{R}R(\overline{X_{0}}+P\Delta X_{0})+P_{R}B_{1}\in\mathbb{R}^{M\times N} \tag{4.16}\]

has rank \(Q<M\) (due to \(P_{R}\)). However, this implies that the \(QM\) unknown matrix components of \(W_{2}\in\mathbb{R}^{Q\times M}\) are determined by \(QN>QM\) equations. This is an overdetermined problem which generically has no solution. However, we can solve the problem to zeroth order in \(\Delta X_{0}\),

\[W_{2}S_{0}=Y^{ext} \tag{4.17}\]

with

\[S_{0}:=S|_{\Delta X_{0}=0}=P_{R}R\overline{X_{0}}=RP\overline{X_{0}} \tag{4.18}\]

(recalling from (3.4) that \(\overline{X_{0}}\) only contains \(Q\) distinct column vectors \(\overline{x_{0,j}}\) and their identical copies) and subsequently minimize the resulting expression in the first term on the r.h.s. of (4.12) with respect to \(P_{R}B_{1}\).

This is equivalent to requiring that \(W_{2}\in\mathbb{R}^{Q\times M}\) solves

\[W_{2}RP\overline{X_{0}^{red}}=Y\,. \tag{4.19}\]

To this end, we make the ansatz

\[W_{2}=A(R\overline{X_{0}^{red}})^{T} \tag{4.20}\]

for some \(A\in\mathbb{R}^{Q\times Q}\). Then, recalling that \(P\overline{X_{0}^{red}}=\overline{X_{0}^{red}}\),

\[A(R\overline{X_{0}^{red}})^{T}(R\overline{X_{0}^{red}})=A(\overline{X_{0}^{red} })^{T}(\overline{X_{0}^{red}})=Y\,, \tag{4.21}\]

so that solving for \(A\), we get

\[W_{2} = Y((\overline{X_{0}^{red}})^{T}\overline{X_{0}^{red}})^{-1}(R \overline{X_{0}^{red}})^{T} \tag{4.22}\] \[= \widetilde{W}_{2}R^{T}\]

with \(\widetilde{W}_{2}\) as defined in (3.38). Hereby, \(W_{2}\) is fully determined by \(Y\) and \(\overline{X_{0}^{red}}\) (as \(\overline{X_{0}^{red}}\) determines \(P\) and \(R\)).
Next, we can choose \(P_{R}^{\perp}b_{1}\) in a suitable manner that

\[\sigma(RP^{\perp}\Delta X_{0}+P_{R}^{\perp}B_{1})=0\,. \tag{4.23}\]

This can be accomplished with

\[P_{R}^{\perp}b_{1}=-\delta P_{R}^{\perp}u_{M}\in\mathbb{R}^{M} \tag{4.24}\]

where \(\delta\) is defined in (3.8), and

\[u_{M}:=(1,1,\ldots,1)^{T}\in\mathbb{R}^{M}\,. \tag{4.25}\]

With

\[E:=u_{M}u_{N}^{T}=[u_{M}\cdots u_{M}]\in\mathbb{R}^{M\times N}\,, \tag{4.26}\]

this ensures that the argument on the l.h.s. of (4.23), given by

\[RP^{\perp}\Delta X_{0}+P_{R}^{\perp}B_{1}=P_{R}^{\perp}R\Delta X_{0}-\delta P _{R}^{\perp}E\,, \tag{4.27}\]

is a matrix all of whose components are \(\leq 0\). Therefore, \(\sigma\) maps it to \(0\).

Consequently, we find

\[\sqrt{N}\mathcal{C}[W_{j},b_{j}]\leq\|W_{2}(RP\Delta X_{0}+P_{R}B_{1})+B_{2}\| _{\mathcal{L}^{2}}\,. \tag{4.28}\]

To minimize the r.h.s., we may shift

\[b_{2}\to b_{2}-W_{2}P_{R}b_{1} \tag{4.29}\]

so that it remains to minimize the r.h.s. of

\[\sqrt{N}\mathcal{C}[W_{j},b_{j}]\leq\|W_{2}RP\Delta X_{0}+B_{2}\|_{\mathcal{L} ^{2}}\,. \tag{4.30}\]

with respect to

\[B_{2}=[b_{2}\cdots b_{2}]=b_{2}u_{N}^{T}\ \in\mathbb{R}^{Q\times N}\,. \tag{4.31}\]

We find

\[0 = \partial_{b_{2}}\|W_{2}RP\Delta X_{0}+\beta_{2}u_{N}^{T}\|_{ \mathcal{L}^{2}}^{2} \tag{4.32}\] \[= \partial_{b_{2}}\Big{(}2\mathrm{Tr}(W_{2}RP\underbrace{\Delta X_ {0}u_{N}}_{=0}b_{2}^{T})+\mathrm{Tr}(u_{N}\underbrace{b_{2}^{T}b_{2}}_{=|b_{2 }|^{2}}u_{N}^{T})\Big{)}\,,\]

using cyclicity of the trace. The first term inside the bracket on the second line is zero because for every \(i\in\{1,\ldots,M\}\),

\[(\Delta X_{0}u_{N})_{i} = \sum_{\ell=1}^{N}[\Delta X_{0}]_{i\ell} \tag{4.33}\] \[= \sum_{j=1}^{Q}\sum_{\ell=1}^{N_{j}}(x_{0,j,\ell}-\overline{x_{0, j}})_{i}\] \[= 0\,.\]

The second term inside the bracket in (4.32) is proportional to \(|b_{2}|^{2}\), so that its gradient is proportional to \(b_{2}\), and vanishes for \(b_{2}=0\). Hence, \(B_{2}=0\), which means that

\[B_{2}=-W_{2}P_{R}B_{1} \tag{4.34}\]

for the original variable \(b_{2}\) prior to the shift (5.10).
We thus arrive at an upper bound

\[\min_{W_{j},b_{j}}\sqrt{N}{\mathcal{C}}[W_{j},b_{j}] \leq \|\widetilde{W}_{2}P\Delta X_{0}\|_{{\mathcal{L}}^{2}} \tag{4.35}\] \[= \|Y{\rm Pen}[\overline{X_{0}^{red}}]P\Delta X_{0}\|_{{\mathcal{L} }^{2}}\,.\]

This establishes the asserted upper bound in (3.19).

Next, we bound

\[\|\widetilde{W}_{2}P\Delta X_{0}\|_{{\mathcal{L}}^{2}} \leq \|Y\|_{op}\|{\rm Pen}[\overline{X_{0}^{red}}]P\Delta X_{0}\|_{{ \mathcal{L}}^{2}}\,. \tag{4.36}\]

Let for notational brevity \(A:={\rm Pen}[\overline{X_{0}^{red}}]P\Delta X_{0}\). We have

\[\|{\rm Pen}[\overline{X_{0}^{red}}]P\Delta X_{0}\|_{{\mathcal{L}} ^{2}}^{2} = {\rm Tr}(AA^{T}) \tag{4.37}\] \[= \sum_{\ell=1}^{Q}\sum_{i=1}^{N_{j}}|{\rm Pen}[\overline{X_{0}^{ red}}]P\Delta x_{0,\ell,i}|^{2}\] \[\leq \Big{(}\sup_{i,\ell}\Big{|}{\rm Pen}[\overline{X_{0}^{red}}]P \Delta x_{0,\ell,i}\Big{|}^{2}\Big{)}\sum_{\ell=1}^{Q}\sum_{i=1}^{N_{\ell}}1\] \[= N\sup_{i,\ell}\Big{|}{\rm Pen}[\overline{X_{0}^{red}}]P\Delta x _{0,\ell,i}\Big{|}^{2}\,\,.\]

Therefore,

\[\min_{W_{j},b_{j}}\sqrt{N}{\mathcal{C}}[W_{j},b_{j}] \leq \|\widetilde{W}_{2}P\Delta X_{0}\|_{{\mathcal{L}}^{2}} \tag{4.38}\] \[\leq \sqrt{N}\,\|Y\|_{op}\sup_{i,\ell}\Big{|}{\rm Pen}[\overline{X_{0 }^{red}}]P\Delta x_{0,\ell,i}\Big{|}\] \[\leq \sqrt{N}\,\|Y\|_{op}\delta_{P}\,,\]

which proves (3.20). This concludes the proof of Theorem 3.1. 

## 5. Proof of Theorem 3.2

To prove the exact degenerate local minimum in the special case \(M=Q\), we note that in this situation,

\[P={\bf 1}_{Q\times Q}\ \ \,,\ P^{\perp}=0\,. \tag{5.1}\]

In place of (4.12), the cost function is given by

\[{\mathcal{C}}_{\mathcal{N}}[W_{j},b_{j}] = \|W_{2}\sigma(W_{1}(\overline{X_{0}}+\Delta X_{0})+B_{1})+B_{2}-Y ^{ext}\|_{{\mathcal{L}}_{\mathcal{N}}^{2}}\,, \tag{5.2}\]

defined in (3.26), which is weighted by the inverse of the block diagonal matrix

\[{\mathcal{N}}={\rm diag}(N_{j}{\bf 1}_{N_{j}\times N_{j}}\,|\,j=1,\ldots,Q)\ \in{ \mathbb{R}}^{N\times N}\,. \tag{5.3}\]

Using a similar choice for \(B_{1}\) as in (4.4), we may assume that the matrix components of \(W_{1}(\overline{X_{0}}+\Delta X_{0})+B_{1}\) are non-negative. To be precise, we let

\[b_{1}=\beta_{1}u_{Q} \tag{5.4}\]

for some \(\beta_{1}\geq 0\) large enough that

\[x_{0,j,i}+\beta_{1}u_{Q}\in{\mathbb{R}}_{+}^{Q} \tag{5.5}\]
for all \(j=1,\ldots,Q\), \(i=1,\ldots,N_{j}\). This is achieved with

\[\beta_{1}\geq 2\max_{j,i}|x_{0,j,i}|\,. \tag{5.6}\]

Hence, choosing

\[W_{1}=\mathbf{1}_{Q\times Q}\,, \tag{5.7}\]

we find

\[\sigma(W_{1}X_{0}+B_{1})=\overline{X_{0}}+\Delta X_{0}+B_{1}\,. \tag{5.8}\]

Thus, 5.2 implies

\[\min_{W_{j},b_{j}}\mathcal{C}_{\mathcal{N}}[W_{j},b_{j}]\leq\|W_{2}(\overline {X_{0}}+\Delta X_{0})+W_{2}B_{1}+B_{2}-Y^{ext}\|_{\mathcal{L}_{\mathcal{N}}^{ 2}}\,. \tag{5.9}\]

To minimize the r.h.s., we note that since \(W_{2}\) and \(B_{2}\) are both unknown, we can redefine \(W_{2}\to W_{2}\), and we can shift

\[b_{2}\to b_{2}-W_{2}b_{1}\,. \tag{5.10}\]

Therefore, we find

\[\min_{W_{j},b_{j}}\mathcal{C}_{\mathcal{N}}[W_{j},b_{j}]\leq\min_{W_{2},b_{2} }\|W_{2}(\overline{X_{0}}+\Delta X_{0})+B_{2}-Y^{ext}\|_{\mathcal{L}_{ \mathcal{N}}^{2}}\,. \tag{5.11}\]

Minimizing with respect to \(W_{2}\in\mathbb{R}^{Q\times Q}\) is a least squares problem whose solution has the form

\[(W_{2}(\overline{X_{0}}+\Delta X_{0})-Y^{ext}_{b_{2}})\mathcal{P}=0\,, \tag{5.12}\]

where

\[Y^{ext}_{b_{2}}:=Y^{ext}-B_{2}\,, \tag{5.13}\]

\(\mathcal{P}^{T}\) is a projector onto the range of \(X_{0}^{T}\), and remains to be determined. To solve (5.12), we require \(W_{2}\) to satisfy

\[W_{2}(\overline{X_{0}}+\Delta X_{0}) \mathcal{N}^{-1}(\overline{X_{0}}+\Delta X_{0})^{T}\] \[=Y^{ext}_{b_{2}}\,\mathcal{N}^{-1}(\overline{X_{0}}+\Delta X_{0} )^{T}\ \ \in\mathbb{R}^{Q\times Q}\,. \tag{5.14}\]

Because

\[\overline{X_{0}} = [\overline{x_{0,1}}u_{N_{1}}^{T}\cdots\overline{x_{0,Q}}u_{N_{Q} }^{T}]\,,\] \[Y^{ext}_{b_{2}} = [(y_{1}-b_{2})u_{N_{1}}^{T}\cdots(y_{Q}-b_{2})u_{N_{Q}}^{T}]\,, \tag{5.15}\]

where

\[u_{N_{j}}:=(1,1,\ldots,1)^{T}\ \ \in\mathbb{R}^{N_{j}}\,, \tag{5.16}\]

we find that

\[\Delta X_{0}\mathcal{N}^{-1}(\overline{X_{0}})^{T} = \sum_{j=1}^{Q}\frac{1}{N_{j}}\Delta X_{0,j}u_{N_{j}}\overline{x_{0,j}}^{T} \tag{5.17}\] \[= \sum_{j=1}^{Q}\underbrace{\Big{(}\frac{1}{N_{j}}\sum_{i=1}^{N_{j }}\Delta x_{0,j,i}\Big{)}}_{=0}\overline{x_{0,j}}^{T}\] \[= 0\,.\]
For the same reason,

\[\overline{X_{0}}{\mathcal{N}}^{-1}\Delta X_{0}^{T}=0\ \ \,\ Y^{ext}{\mathcal{N}}^{-1} \Delta X_{0}^{T}=0\,. \tag{5.18}\]

Moreover,

\[Y_{b_{2}}^{ext}{\mathcal{N}}^{-1}\overline{X_{0}}^{T} = \sum_{j=1}^{Q}\frac{1}{N_{j}}(y_{j}-b_{2})\underbrace{u_{N_{j}}^{ T}u_{N_{j}}}_{=N_{j}}\overline{x_{0,j}}^{T} \tag{5.19}\] \[= Y_{b_{2}}\overline{X_{0}^{red}}^{T}\,,\]

where

\[Y_{b_{2}}=[(y_{1}-b_{2})\cdots(y_{Q}-b_{2})]\ \ \in{\mathbb{R}}^{Q\times Q} \tag{5.20}\]

and

\[\overline{X_{0}}{\mathcal{N}}^{-1}\overline{X_{0}}^{T}=\overline{X_{0}^{red} \,X_{0}^{red}}^{T}\,. \tag{5.21}\]

Therefore, (5.14) reduces to

\[W_{2}(\overline{X_{0}^{red}}^{T}\overline{X_{0}^{red}}^{T}+\Delta X_{0}{ \mathcal{N}}^{-1}\Delta X_{0}^{T})=Y_{b_{2}}\overline{X_{0}^{red}}^{T}\,, \tag{5.22}\]

and invertibility of \(\overline{X_{0}^{red}}\overline{X_{0}^{red}}^{T}\in{\mathbb{R}}^{Q\times Q}\) implies invertibility of the matrix in brackets on the l.h.s. of (5.22), because

\[D_{2}[X_{0}]:=\Delta X_{0}{\mathcal{N}}^{-1}\Delta X_{0}^{T}=\sum_{j=1}^{Q} \frac{1}{N_{j}}\sum_{i=1}^{N_{j}}\Delta x_{0,j,i}\Delta x_{0,j,i}^{T}\ \ \in{\mathbb{R}}^{Q\times Q} \tag{5.23}\]

is a non-negative operator. Hence,

\[W_{2}=Y_{b_{2}}\overline{X_{0}^{red}}^{T}(\overline{X_{0}^{red}\,X_{0}^{red}} ^{T}+\Delta X_{0}{\mathcal{N}}^{-1}\Delta X_{0}^{T})^{-1}\,, \tag{5.24}\]

so that

\[W_{2}X_{0}=Y_{b_{2}}^{ext}{\mathcal{P}}\,. \tag{5.25}\]

Here,

\[{\mathcal{P}}:={\mathcal{N}}^{-1}X_{0}^{T}(X_{0}{\mathcal{N}}^{-1}X_{0}^{T})^ {-1}X_{0}\in{\mathbb{R}}^{N\times N} \tag{5.26}\]

is a rank \(Q\) projector \({\mathcal{P}}^{2}={\mathcal{P}}\) which is orthogonal with respect to the inner product on \({\mathbb{R}}^{N}\) defined by \({\mathcal{N}}\), i.e., \({\mathcal{P}}^{T}{\mathcal{N}}={\mathcal{N}}{\mathcal{P}}\). \({\mathcal{P}}^{T}\) is the projector onto the range of \(X_{0}^{T}\). We note that with respect to the inner product defined with \({\mathcal{N}}^{-1}\) (which we are using here), we have \({\mathcal{N}}^{-1}{\mathcal{P}}^{T}={\mathcal{P}}{\mathcal{N}}^{-1}\).

Next, in order to control \(W_{2}X_{0}-Y_{b_{2}}^{ext}=-Y_{b_{2}}^{ext}{\mathcal{P}}^{\perp}\), we observe that

\[Y_{b_{2}}^{ext}{\mathcal{P}} = Y_{b_{2}}^{ext}{\mathcal{N}}^{-1}X_{0}^{T}(X_{0}{\mathcal{N}}^{ -1}X_{0}^{T})^{-1}X_{0} \tag{5.27}\] \[= Y_{b_{2}}\overline{X_{0}^{red}}^{T}(\overline{X_{0}^{red}\,X_{0} ^{red}}^{T}+D_{2}[X_{0}])^{-1}X_{0}\] \[= Y_{b_{2}}\overline{X_{0}^{red}}^{T}(\overline{X_{0}^{red}\,X_{0} ^{red}}^{T})^{-1}X_{0}\] \[-Y_{b_{2}}\overline{X_{0}^{red}}^{T}(\overline{X_{0}^{red}\,X_{0} ^{red}}^{T})^{-1}D_{2}[X_{0}](X_{0}{\mathcal{N}}^{-1}X_{0}^{T})^{-1}X_{0}\] \[= Y_{b_{2}}(\overline{X_{0}^{red}})^{-1}X_{0}\] \[-Y_{b_{2}}(\overline{X_{0}^{red}})^{-1}D_{2}[X_{0}](X_{0}{ \mathcal{N}}^{-1}X_{0}^{T})^{-1}X_{0}\]
where we have used the matrix identity

\[(A+B)^{-1} = A^{-1}-A^{-1}B(A+B)^{-1} \tag{5.28}\] \[= A^{-1}-(A+B)^{-1}BA^{-1}\]

for \(A\), \(A+B\) invertible. We observe that

\[Y_{b_{2}}(\overline{X_{0}^{red}})^{-1}X_{0} = Y_{b_{2}}(\overline{X_{0}^{red}})^{-1}\overline{X_{0}}+Y_{b_{2} }(\overline{X_{0}^{red}})^{-1}\Delta X_{0} \tag{5.29}\] \[= Y_{b_{2}}^{ext}+Y_{b_{2}}(\overline{X_{0}^{red}})^{-1}\Delta X_ {0}\,.\]

To pass to the last line, we used that \((\overline{X_{0}^{red}})^{-1}\overline{X_{0}}=[e_{1}u_{N_{1}}^{T}\cdots e_{Q}u _{N_{Q}}^{T}]\) where \(\{e_{j}\in\mathbb{R}^{Q}\,|\,j=1,\ldots,Q\}\) are the unit basis vectors.

We therefore conclude that, due to \(\mathcal{P}^{\perp}=\mathbf{1}-\mathcal{P}\),

\[Y_{b_{2}}^{ext}\mathcal{P}^{\perp} = -Y_{b_{2}}(\overline{X_{0}^{red}})^{-1}\Delta X_{0}+\mathcal{R} \tag{5.30}\]

where

\[\mathcal{R} := Y_{b_{2}}(\overline{X_{0}^{red}})^{-1}D_{2}[X_{0}](X_{0} \mathcal{N}^{-1}X_{0}^{T})^{-1}X_{0} \tag{5.31}\]

satisfies

\[\mathcal{RP}=\mathcal{R}\,, \tag{5.32}\]

as can be easily verified.

At this point, we have found that

\[\min_{W_{j},b_{j}}\mathcal{C}_{\mathcal{N}}[W_{j},b_{j}]\leq\min_{b_{2}}\|Y_{b _{2}}^{ext}\mathcal{P}^{\perp}\|_{\mathcal{L}_{\mathcal{N}}^{2}}\,. \tag{5.33}\]

To minimize with respect to \(b_{2}\), we set

\[0 = \partial_{b_{2}}\|Y_{b_{2}}^{ext}\mathcal{P}^{\perp}\|_{\mathcal{ L}_{\mathcal{N}}^{2}}^{2} \tag{5.34}\] \[= \partial_{b_{2}}\Big{(}-2\mathrm{Tr}\Big{(}Y^{ext}\mathcal{P}^{ \perp}\mathcal{N}^{-1}(\mathcal{P}^{\perp})^{T}u_{N}b_{2}^{T}\Big{)}+\mathrm{ Tr}\big{(}b_{2}u_{N}^{T}\mathcal{P}^{\perp}\mathcal{N}^{-1}(\mathcal{P}^{\perp})^{T}u_{N} b_{2}^{T}\big{)}\Big{)}\] \[= \partial_{b_{2}}\Big{(}-2\mathrm{Tr}\Big{(}b_{2}^{T}Y^{ext} \mathcal{P}^{\perp}\mathcal{N}^{-1}(\mathcal{P}^{\perp})^{T}u_{N}\Big{)}+| \mathcal{N}^{-1/2}(\mathcal{P}^{\perp})^{T}u_{N}|^{2}|b_{2}|^{2}\Big{)}\] \[= \partial_{b_{2}}\Big{(}-2\mathrm{Tr}\Big{(}b_{2}^{T}Y_{b_{2}}( \overline{X_{0}^{red}})^{-1}\underbrace{\Delta X_{0}\mathcal{N}^{-1}u_{N}}_{=0 }\Big{)}\ +\ \mathrm{Tr}\Big{(}b_{2}^{T}\underbrace{\mathcal{R}\mathcal{N}^{-1}(\mathcal{P}^ {\perp})^{T}}_{=\mathcal{RP}\mathcal{N}^{-1}\mathcal{P}^{\perp}=0}u_{N}\Big{)}\] \[+|\mathcal{N}^{-1/2}(\mathcal{P}^{\perp})^{T}u_{N}|^{2}|b_{2}|^{2}\Big{)}\]

using \(\mathcal{R}=\mathcal{RP}\) from (5.32) so that \(\mathcal{R}\mathcal{N}^{-1}\mathcal{P}^{\perp}=\mathcal{RP}\mathcal{N}^{-1} \mathcal{P}^{\perp}=\mathcal{RP}\mathcal{P}^{\perp}\mathcal{N}^{-1}=0\), and recalling \(u_{N}=(1,1,\ldots,1)^{T}\in\mathbb{R}^{N}\). This implies that

\[b_{2}=0\,. \tag{5.35}\]

We note that this corresponds to

\[b_{2}=-W_{2}b_{1} \tag{5.36}\]

prior to the shift (5.10), and hence arrive at (3.33), as claimed.

Next, we derive a simplified expression for \(\|Y^{ext}\mathcal{P}^{\perp}\|_{\mathcal{L}_{\mathcal{N}}^{2}}^{2}\). For notational convenience, let

\[\Delta_{1}^{rel} := (\overline{X_{0}^{red}})^{-1}\Delta X_{0}\] \[\Delta_{2}^{rel} := (\overline{X_{0}^{red}})^{-1}D_{2}[X_{0}](\overline{X_{0}^{red}})^ {-T} \tag{5.37}\] \[= \Delta_{1}^{rel}\mathcal{N}^{-1}(\Delta_{1}^{rel})^{T}\]
where we recall (5.23). Then, we obtain (with \(b_{2}=0\))

\[\mathcal{R} = Y\Delta_{2}^{rel}\big{(}1+\Delta_{2}^{rel}\big{)}^{-1}(\overline{X _{0}^{red}})^{-1}X_{0} \tag{5.38}\] \[= Y\Delta_{2}^{rel}\big{(}1+\Delta_{2}^{rel}\big{)}^{-1}\Big{(}[e_{ 1}u_{N_{1}}^{T}\cdots e_{Q}u_{N_{Q}}^{T}]+\Delta_{1}^{rel}\Big{)}\]

where we used \(X_{0}\mathcal{N}^{-1}X_{0}^{T}=\overline{X_{0}^{red}}(1+\Delta_{2}^{rel})( \overline{X_{0}^{red}})^{T}\) in (5.31) to obtain the first line. We find that

\[-Y^{ext}\mathcal{P}^{\perp} \tag{5.39}\] \[= Y\Delta_{1}^{rel}-Y\Delta_{2}^{rel}\big{(}1+\Delta_{2}^{rel} \big{)}^{-1}\Big{(}[e_{1}u_{N_{1}}^{T}\cdots e_{Q}u_{N_{Q}}^{T}]+\Delta_{1}^{ rel}\Big{)}\] \[= Y(1-\Delta_{2}^{rel}\big{(}1+\Delta_{2}^{rel}\big{)}^{-1})\Delta _{1}^{rel}-Y\Delta_{2}^{rel}\big{(}1+\Delta_{2}^{rel}\big{)}^{-1}[e_{1}u_{N_{1 }}^{T}\cdots e_{Q}u_{N_{Q}}^{T}]\] \[= Y(1+\Delta_{2}^{rel})^{-1}\Delta_{1}^{rel}-Y\Delta_{2}^{rel} \big{(}1+\Delta_{2}^{rel}\big{)}^{-1}[e_{1}u_{N_{1}}^{T}\cdots e_{Q}u_{N_{Q}}^ {T}]\,.\]

Therefore, we obtain that

\[\|Y^{ext}\mathcal{P}^{\perp}\|_{\mathcal{L}_{\mathcal{N}}^{2}}^{2} = (I)+(II)+(III) \tag{5.40}\]

where

\[(I) := \mbox{Tr}\Big{(}Y(1+\Delta_{2}^{rel})^{-1}\Delta_{1}^{rel} \mathcal{N}^{-1}(\Delta_{1}^{rel})^{T}(1+\Delta_{2}^{rel})^{-T}Y^{T}\Big{)} \tag{5.41}\] \[= \mbox{Tr}\Big{(}Y(1+\Delta_{2}^{rel})^{-1}\Delta_{2}^{rel}(1+ \Delta_{2}^{rel})^{-1}Y^{T}\Big{)}\]

where we note that \(\Delta_{2}^{rel}=(\Delta_{2}^{rel})^{T}\in\mathbb{R}^{Q\times Q}\) is symmetric. Moreover,

\[(II) := -2\mbox{Tr}\Big{(}Y(1+\Delta_{2}^{rel})^{-1}\Delta_{1}^{rel} \mathcal{N}^{-1}[e_{1}u_{N_{1}}^{T}\cdots e_{Q}u_{N_{Q}}^{T}]^{T}\big{(}1+ \Delta_{2}^{rel}\big{)}^{-1}\Delta_{2}^{rel}Y^{T}\Big{)}\] \[= -\sum_{j=1}^{Q}2\mbox{Tr}\Big{(}Y(1+\Delta_{2}^{rel})^{-1}( \overline{X_{0}^{red}})^{-1}\underbrace{\frac{1}{N_{j}}\Delta X_{0,j}u_{N_{j}} e_{j}^{T}}_{=0}\big{(}1+\Delta_{2}^{rel}\big{)}^{-1}\Delta_{2}^{rel}Y^{T}\Big{)}\] \[= 0\]

where we used that \(\frac{1}{N_{j}}\Delta X_{0,j}u_{N_{j}}=\overline{\Delta X_{0,j}}=0\) for all \(j=1,\ldots,Q\). Furthermore,

\[(III) := \mbox{Tr}\Big{(}Y\Delta_{2}^{rel}\big{(}1+\Delta_{2}^{rel}\big{)} ^{-1}[e_{1}u_{N_{1}}^{T}\cdots e_{Q}u_{N_{Q}}^{T}]\mathcal{N}^{-1}[e_{1}u_{N_ {1}}^{T}\cdots e_{Q}u_{N_{Q}}^{T}]^{T} \tag{5.43}\] \[\qquad\qquad\qquad\big{(}1+\Delta_{2}^{rel}\big{)}^{-1}\Delta_{2} ^{rel}Y^{T}\Big{)}\] \[= \mbox{Tr}\Big{(}Y\Delta_{2}^{rel}\big{(}1+\Delta_{2}^{rel}\big{)} ^{-1}\sum_{j=1}^{Q}\underbrace{\frac{1}{N_{j}}u_{N_{j}}^{T}u_{N_{j}}}_{=1}e_{ j}e_{j}^{T}\big{(}1+\Delta_{2}^{rel}\big{)}^{-1}\Delta_{2}^{rel}Y^{T}\Big{)}\] \[= \mbox{Tr}\Big{(}Y\Delta_{2}^{rel}\big{(}1+\Delta_{2}^{rel}\big{)} ^{-1}\big{(}1+\Delta_{2}^{rel}\big{)}^{-1}\Delta_{2}^{rel}Y^{T}\Big{)}\] \[= \mbox{Tr}\Big{(}Y\big{(}1+\Delta_{2}^{rel}\big{)}^{-1}(\Delta_{2} ^{rel})^{2}\big{(}1+\Delta_{2}^{rel}\big{)}^{-1}Y^{T}\Big{)}\]

using \(\sum_{j=1}^{Q}e_{j}e_{j}^{T}=\mathbf{1}_{Q\times Q}\), and commutativity of \(\Delta_{2}^{rel}\) and \((1+\Delta_{2}^{rel})^{-1}\).
We conclude that

\[\|Y^{ext}{\mathcal{P}}^{\perp}\|^{2}_{{\mathcal{K}}_{N}} = (I)+(III) \tag{5.44}\] \[= {\rm Tr}\Big{(}Y\big{(}1+\Delta_{2}^{rel}\big{)}^{-1}\Delta_{2}^{ rel}(1+\Delta_{2}^{rel})\big{(}1+\Delta_{2}^{rel}\big{)}^{-1}Y^{T}\Big{)}\] \[= {\rm Tr}\Big{(}Y\Delta_{2}^{rel}\big{(}1+\Delta_{2}^{rel}\big{)}^{ -1}Y^{T}\Big{)}\] \[= {\rm Tr}\Big{(}Y|\Delta_{2}^{rel}|^{\frac{1}{2}}\big{(}1+\Delta_{2 }^{rel}\big{)}^{-1}|\Delta_{2}^{rel}|^{\frac{1}{2}}Y^{T}\Big{)}\] \[\leq \|\big{(}1+\Delta_{2}^{rel}\big{)}^{-1}\|_{op}\;{\rm Tr}\Big{(}Y \Delta_{2}^{rel}Y^{T}\Big{)}\]

since the matrix \(\Delta_{2}^{rel}\geq 0\) is positive semidefinite. We note that

\[{\rm Tr}\Big{(}Y\Delta_{2}^{rel}Y^{T}\Big{)} = {\rm Tr}\Big{(}Y\Delta_{1}^{rel}{\mathcal{N}}^{-1}(\Delta_{1}^{ rel})^{T}Y^{T}\Big{)} \tag{5.45}\] \[= \|Y(\overline{X_{0}^{red}})^{-1}\Delta X_{0}\|^{2}_{{\mathcal{L} }_{N}^{2}}\,.\]

Moreover, letting \(\lambda_{-}:=\inf{\rm spec}(\Delta_{2}^{rel})\) and \(\lambda_{+}:=\sup{\rm spec}(\Delta_{2}^{rel})\) denote the smallest and largest eigenvalue of the non-negative, symmetric matrix \(\Delta_{2}^{rel}\geq 0\),

\[0\leq\lambda_{-}\leq\lambda_{+}=\|\Delta_{2}^{rel}\|_{op}=\|\Delta_{1}^{rel}{ \mathcal{N}}^{-1}(\Delta_{1}^{rel})^{T}\|_{op}\leq C\delta_{P}^{2}\,. \tag{5.46}\]

Therefore,

\[\|\big{(}1+\Delta_{2}^{rel}\big{)}^{-1}\|_{op}=(1+\lambda_{-})^{-1}\leq 1-C_ {0}^{\prime}\delta_{P}^{2} \tag{5.47}\]

for a constant \(C_{0}^{\prime}=0\) if \({\rm rank}(\Delta_{2}^{rel})<Q\) whereby \(\lambda_{-}=0\), and \(C_{0}^{\prime}>0\) if \({\rm rank}(\Delta_{2}^{rel})=Q\).

We find the following bound from (5.44),

\[\|Y^{ext}{\mathcal{P}}^{\perp}\|_{{\mathcal{L}}_{N}^{2}} \leq \sqrt{\|\big{(}1+\Delta_{2}^{rel}\big{)}^{-1}\|_{op}\;{\rm Tr} \Big{(}Y\Delta_{2}^{rel}Y^{T}\Big{)}} \tag{5.48}\] \[\leq (1-C_{0}\delta_{P}^{2})\;\|Y(\overline{X_{0}^{red}})^{-1}\Delta X _{0}\|_{{\mathcal{L}}_{N}^{2}}\,,\]

as claimed, where \(C_{0}\) is proportional to \(C_{0}^{\prime}\).

Finally, we use that

\[\widetilde{W}_{2} = Y\overline{X_{0}^{red}}^{T}(\overline{X_{0}^{red}}\,\overline{X _{0}^{red}}^{T})^{-1} \tag{5.49}\] \[= Y(\overline{X_{0}^{red}})^{-1}\]

agrees with (3.38) because \(\overline{X_{0}^{red}}\) is invertible, and hence \((\overline{X_{0}^{red}})^{-1}={\rm Pen}[\overline{X_{0}^{red}}]\). Hence, we observe that (5.24) implies

\[W_{2} = \widetilde{W}_{2}-\widetilde{W}_{2}(\Delta X_{0}{\mathcal{N}}^{- 1}\Delta X_{0}^{T})(\overline{X_{0}^{red}}\,\overline{X_{0}^{red}}^{T}+\Delta X _{0}{\mathcal{N}}^{-1}\Delta X_{0}^{T})^{-1} \tag{5.50}\] \[= \widetilde{W}_{2}-Y(\overline{X_{0}^{red}})^{-1}(\Delta X_{0}{ \mathcal{N}}^{-1}\Delta X_{0}^{T})(\overline{X_{0}^{red}})^{-T}\] \[\qquad\qquad\quad\Big{(}{\bf 1}+(\overline{X_{0}^{red}})^{-1}( \Delta X_{0}{\mathcal{N}}^{-1}\Delta X_{0}^{T})(\overline{X_{0}^{red}})^{-T} \Big{)}^{-1}(\overline{X_{0}^{red}})^{-1}\] \[= \widetilde{W}_{2}-Y\Delta_{2}^{rel}({\bf 1}+\Delta_{2}^{rel})^{-1}( \overline{X_{0}^{red}})^{-1}\]

Therefore, in operator norm,

\[\|W_{2}-\widetilde{W}_{2}\|_{op} \leq \|Y\|_{op}\|\Delta_{2}^{rel}\|_{op}\|({\bf 1}+\Delta_{2}^{rel})^{-1}\|_{op} \|(\overline{X_{0}^{red}})^{-1}\|_{op} \tag{5.51}\] \[\leq C\delta_{P}^{2}\]
where we used \(\|Y\|_{op}<C\), \(\|\Delta_{2}^{rel}\|_{op}<C\widetilde{\delta}_{P}^{2}\), \(\|(\mathbf{1}+\Delta_{2}^{rel})^{-1}\|_{op}\leq 1\), \(\|(\overline{X_{0}^{red}})^{-1}\|_{op}<C\) to obtain the last line.

We note that when \(M>Q\), it follows that \(\overline{X_{0}^{red}}(\overline{X_{0}^{red}})^{T}\) is an \(M\times M\) matrix of rank \(Q\), and is hence not invertible, so that the above arguments do not apply.

From here on, we denote the weights and shifts determined above in (5.7), (5.24) and in (5.4), (5.36), by \(W_{1}^{*},W_{2}^{*}\) and \(b_{1}^{*},b_{2}^{*}\), respectively. To prove that \(\mathcal{C}_{\mathcal{N}}[W_{i}^{*},b_{i}^{*}]\) is a local minimum of the cost function \(\mathcal{C}_{\mathcal{N}}\), we observe that for

\[\rho:=\max_{j,i}|x_{0,j,i}| \tag{5.52}\]

and

\[\beta_{1}\geq 2\rho\ \,\ \ b_{1}^{*}=\beta_{1}u_{Q}\,, \tag{5.53}\]

we have

\[W_{1}^{*}x_{0,j,i}+b_{1}^{*} = x_{0,j,i}+b_{1}^{*} \tag{5.54}\] \[\in B_{\rho}(\beta_{1}u_{Q})\subset\mathbb{R}_{+}^{Q}\,.\]

\(\forall j=1,\ldots,Q\) and \(\forall\;i=1,\ldots,N_{j}\). That is, all training input vectors are contained in a ball of radius \(\rho\) centered at a point on the diagonal \(\beta_{1}u_{Q}\in\mathbb{R}^{Q}\) with \(\beta_{1}\geq 2\rho\). This ensures that the coordinates of all \(W_{1}^{*}x_{0,j,i}+b_{1}^{*}\) are strictly positive, and therefore,

\[\sigma(W_{1}^{*}X_{0}+B_{1}^{*})=W_{1}^{*}X_{0}+B_{1}^{*}\,. \tag{5.55}\]

For \(\epsilon>0\) small, we consider an infinitesimal translation

\[b_{1}^{*}\to b_{1}^{*}+\widetilde{b}_{1} \tag{5.56}\]

combined with an infinitesimal transformation

\[W_{1}^{*}\to W_{1}^{*}\widetilde{W}_{1} \tag{5.57}\]

where \(|\widetilde{b}_{1}|<\epsilon\) and \(\|\widetilde{W}_{1}-\mathbf{1}_{Q\times Q}\|_{op}<\epsilon\). Then, we have that

\[W_{1}^{*}\widetilde{W}_{1}x_{0,j,i}+b_{1}^{*}+\widetilde{b}_{1} \tag{5.58}\] \[= \underbrace{W_{1}^{*}x_{0,j,i}+b_{1}^{*}}_{\in B_{\rho}(\beta_{1} u_{Q})}+W_{1}^{*}(\widetilde{W}_{1}-\mathbf{1}_{Q\times Q})x_{0,j,i}+\widetilde{b}_{1}\]

where

\[|W_{1}^{*}(\widetilde{W}_{1}-\mathbf{1}_{Q\times Q})x_{0,j,i}+ \widetilde{b}_{1}| \leq \|W_{1}^{*}\|_{op}\|\widetilde{W}_{1}-\mathbf{1}_{Q\times Q}\|_{ op}|x_{0,j,i}|+|\widetilde{b}_{1}| \tag{5.59}\] \[\leq \|W_{1}^{*}\|_{op}\;\epsilon\;\max_{j,i}|x_{0,j,i}|+\epsilon\] \[\leq C\epsilon\]

for some constant \(C\) independent of \(j,i\). Therefore, we conclude that (5.58) is contained in \(B_{\rho+C\epsilon}(\beta_{1}u_{Q})\), which lies in the positive sector \(\mathbb{R}_{+}^{Q}\subset\mathbb{R}^{Q}\) for sufficiently small \(\epsilon>0\). But this implies that

\[\sigma(W_{1}^{*}\widetilde{W}_{1}X_{0}+B_{1}^{*}+\widetilde{B}_{1})=W_{1}^{*} \widetilde{W}_{1}X_{0}+B_{1}^{*}+\widetilde{B}_{1}\,. \tag{5.60}\]

Thus, all arguments leading to the result in (3.29) stating that

\[\mathcal{C}_{\mathcal{N}}[W_{i}^{*},b_{i}^{*}]=\|Y^{ext}\mathcal{P}^{\perp}\|_{ \mathcal{L}_{\mathcal{N}}^{2}}\,, \tag{5.61}\]

for the choice of weights and shifts \(W_{i}^{*},b_{i}^{*}\), equally apply to the infinitesimally deformed \(W_{1}^{*}\widetilde{W}_{1}\), \(b_{1}^{*}+\widetilde{b}_{1}\) and the corresponding expressions for \(W_{2}^{*}\widetilde{W}_{2}\), \(b_{2}^{*}+\widetilde{b}_{2}\).
We now make the crucial observation that the right hand side of (5.61) does not depend on \(W_{i}^{*},b_{i}^{*}\), as \(\mathcal{P}^{\perp}\) only depends on the training inputs \(X_{0}\); see (5.26).

Therefore,

\[\mathcal{C}_{\mathcal{N}}[W_{i}^{*},b_{i}^{*}]=\mathcal{C}_{\mathcal{N}}[W_{i}^ {*}\widetilde{W}_{i},b_{i}^{*}+\widetilde{b}_{i}]\,, \tag{5.62}\]

and the variation in the perturbations \(\widetilde{W}_{i},\widetilde{b}_{i}\) is zero. This implies that \(\mathcal{C}_{\mathcal{N}}[W_{i}^{*},b_{i}^{*}]\) is a local minimum of the cost function, as it was obtained from a least square minimization problem. In particular, it is degenerate, and by repeating the above argument, one concludes that it has the same value for all weights and shifts \(W_{i},b_{i}\) which allow for the condition (5.55) to be satisfied.

Finally, we note that the reparametrization of the training inputs \(X_{0}\to KX_{0}\), for any arbitrary \(K\in GL(Q)\), induces

\[\overline{X_{0}}\to K\overline{X_{0}}, \Delta X_{0}\to K\Delta X_{0}\,. \tag{5.63}\]

In particular, this implies that

\[\Delta_{1}^{rel} = (\overline{X_{0}^{red}})^{-1}\Delta X_{0} \tag{5.64}\] \[\to (K\overline{X_{0}^{red}})^{-1}K\Delta X_{0}\] \[= (\overline{X_{0}^{red}})^{-1}K^{-1}K\Delta X_{0}\] \[= \Delta_{1}^{rel}\]

is invariant under this reparametrization, and hence,

\[\Delta_{2}^{rel} = \Delta_{1}^{rel}\mathcal{N}^{-1}(\Delta_{1}^{rel})^{T} \tag{5.65}\]

also is. From the third and fourth lines in (5.44) follows that \(\mathcal{C}_{\mathcal{N}}[W_{i}^{*},b_{i}^{*}]\) is a function only of \(\Delta_{2}^{rel}\). We note that the reparametrization invariance of \(\mathcal{P}\) (and thus of \(\mathcal{P}^{\perp}\)) can also be directly verified from (5.26) by inspection. This implies that \(\mathcal{C}_{\mathcal{N}}[W_{i}^{*},b_{i}^{*}]\) is invariant under reparametrizations of the training inputs \(X_{0}\to KX_{0}\), for all \(K\in GL(Q)\).

This concludes the proof of Theorem 3.2. 

## 6. Constructive training of shallow network

In this section, we follow the proof of Theorem 3.1 in order to determine a constructively trained shallow network, which does not invoke the gradient descent algorithm.

In our approach, all relevant operations are confined to a \(Q\leq M\) dimensional linear subspace of \(\mathbb{R}^{M}\). To this end, we calculate

\[\overline{X_{0}^{red}}:=[\overline{x_{0,1}}\cdots\overline{x_{0,Q}}]\in \mathbb{R}^{M\times Q}\,. \tag{6.1}\]

We determine the Penrose inverse

\[\mathrm{Pen}[\overline{X_{0}^{red}}]=\big{(}(\overline{X_{0}^{red}})^{T} \overline{X_{0}^{red}}\big{)}^{-1}(\overline{X_{0}^{red}})^{T}\in\mathbb{R}^{Q \times M} \tag{6.2}\]

and the orthoprojector

\[P:=\overline{X_{0}^{red}}\mathrm{Pen}[\overline{X_{0}^{red}}]\in\mathbb{R}^{ M\times M}\,. \tag{6.3}\]

Subsequently, we determine \(W_{1}=R\in O(M)\) which acts by rotating the ranges of \(P\) and \(P^{\perp}\) to be in alignment with the coordinate vectors \(e_{j}:=(0,\ldots 0,1,0,\ldots,0)\). As \(P\) is symmetric, this amounts to solving the diagonalization problem

\[P=R^{T}P_{R}R\,. \tag{6.4}\]
\(P_{R}\) is diagonal, with \(Q\) diagonal entries equaling \(1\), and all other \(M^{2}-Q\) matrix entries being \(0\). The orthogonal matrix \(R\) diagonalizing \(P\) also diagonalizes \(P^{\perp}={\bf 1}-P\). We let

\[W_{1}^{*}=R\in O(M)\,. \tag{6.5}\]

Moreover, we set

\[W_{2}^{*}=\widetilde{W}_{2}R^{T} \tag{6.6}\]

where we calculate

\[\widetilde{W}_{2}=Y((\overline{X_{0}^{red}})^{T}\overline{X_{0}^{red}})^{-1} (\overline{X_{0}^{red}})^{T}\in\mathbb{R}^{Q\times M}\,. \tag{6.7}\]

To determine \(B_{1}^{*}\), we split \(b_{1}^{*}=P_{R}b_{1}^{*}+P_{R}^{\perp}b_{1}^{*}\), and set

\[P_{R}b_{1}^{*}=\beta_{1}P_{R}u_{M}\,, \tag{6.8}\]

recalling \(u_{M}=(1,1,\ldots,1)^{T}\in\mathbb{R}^{M}\) from (4.25). We choose \(\beta_{1}\geq 0\) large enough to ensure that \(RPX_{0}+P_{R}B_{1}^{*}\) is component-wise strictly positive. A sufficient condition is \(\beta_{1}\geq 2\max_{j,i}|x_{0,j,i}|\). Moreover, we set

\[P_{R}^{\perp}b_{1}^{*}=-\delta P_{R}^{\perp}u_{M} \tag{6.9}\]

with \(\delta\) as defined in (3.8). This defines \(B_{1}^{*}=[b_{1}^{*}\cdots b_{1}^{*}]\in\mathbb{R}^{M\times N}\). Finally, we let

\[b_{2}^{*}=-W_{2}^{*}P_{R}b_{1}^{*}\,. \tag{6.10}\]

The set \(W_{1}^{*},W_{2}^{*},b_{1}^{*},b_{2}^{*}\) defines the constructively trained shallow network. No use of the gradient descent algorithm is required.

## 7. Proof of Theorem 3.3

Here, we prove Theorem 3.3, and elucidate the geometric meaning of the constructively trained network obtained in Section 6. Given a non-training input \(x\in\mathbb{R}^{M}\), the constructively trained network metrizes the linear subspace \(\operatorname{range}(P)\) of the input space \(\mathbb{R}^{M}\), and determines to which equivalence class of training inputs \(Px\) is closest. To elucidate this interpretation, we observe that (6.7) implies that

\[\widetilde{W}_{2}P=\widetilde{W}_{2}\quad\text{and}\;W_{2}^{*}P_{R}=W_{2}^{*} \tag{7.1}\]

and hence,

\[\widetilde{W}_{2}\overline{X_{0}^{red}}=Y\,, \tag{7.2}\]

or in terms of column vectors,

\[W_{2}^{*}RP\overline{x_{0,j}}=\widetilde{W}_{2}\overline{x_{0,j}}=y_{j}\,. \tag{7.3}\]

Therefore,

\[\mathcal{C}_{j}[x] = |W_{2}^{*}(W_{1}^{*}x+b_{1}^{*})_{+}+b_{2}^{*}-y_{j}| \tag{7.4}\] \[= |W_{2}^{*}(Rx+b_{1}^{*})_{+}+b_{2}^{*}-y_{j}|\] \[= |W_{2}^{*}(RPx+RP^{\perp}x+P_{R}b_{1}^{*}+P_{R}^{\perp}b_{1}^{*} )_{+}+b_{2}^{*}-y_{j}|\] \[= |W_{2}^{*}((P_{R}RPx+P_{R}b_{1}^{*})_{+}+(P_{R}^{\perp}RP^{\perp }x+P_{R}^{\perp}b_{1}^{*})_{+})+b_{2}^{*}-y_{j}|\] \[= |W_{2}^{*}(P_{R}RPx-y_{j})+W_{2}(P_{R}^{\perp}RP^{\perp}x-\delta P _{R}^{\perp}u)_{+}|\] \[= |W_{2}^{*}(P_{R}RP(x-\overline{x_{0,j}}))|\]

where by construction of \(P\), we have \(P\overline{x_{0,j}}=\overline{x_{0,j}}\). Passing to the fourth line, we used \(P_{R}RP=RP\), passing to the fifth line, we recalled \(W_{2}^{*}P_{R}b_{1}^{*}+b_{2}^{*}=0\), and
passing to the sixth line, we used (7.3). Moreover, we recall that \(u\) is defined in (4.25). To pass to the sixth line, we also used that

\[P_{R}^{\perp}(P_{R}^{\perp}RP^{\perp}x-\delta P_{R}^{\perp}u)_{+}=(P_{R}^{\perp} RP^{\perp}x-\delta P_{R}^{\perp}u)_{+} \tag{7.5}\]

because the vector inside \((\cdot)_{+}\) only has components in the range of \(P_{R}^{\perp}\), which is diagonal in the given coordinate system, and because \((\cdot)_{+}\) acts component-wise. Therefore, since also \(W_{2}^{*}P_{R}=W_{2}^{*}\), we have

\[W_{2}^{*}(P_{R}^{\perp}RP^{\perp}x-\delta P_{R}^{\perp}u)_{+} \tag{7.6}\] \[= W_{2}^{*}P_{R}P_{R}^{\perp}(P_{R}^{\perp}RP^{\perp}x-\delta P_{ R}^{\perp}u)_{+}\] \[= 0\,.\]

With \(W_{2}^{*}P_{R}RP=\widetilde{W}_{2}P\), (7.4) implies that

\[\mathcal{C}_{j}[x]=d_{\widetilde{W}_{2}}(Px,\overline{x_{0,j}}) \tag{7.7}\]

where

\[d_{\widetilde{W}_{2}}(x,y):=|\widetilde{W}_{2}P(x-y)|\,, \tag{7.8}\]

for \(x,y\in\operatorname{range}(P)\), defines a metric in the range of \(P\), which is a \(Q\)-dimensional linear subspace of the input space \(\mathbb{R}^{M}\). This is because the matrix \(\widetilde{W}_{2}=\widetilde{W}_{2}P\) has full rank \(Q\).

We can therefore reformulate the identification of an input \(x\in\mathbb{R}^{M}\) with an output \(y_{j^{*}}\) via the constructively trained shallow network as the solution of the metric minimization problem

\[j^{*}=\operatorname{argmin}_{j\in\{1,\ldots,Q\}}(d_{\widetilde{W}_{2}}(Px, \overline{x_{0,j}})) \tag{7.9}\]

in the range of \(P\). 

**Acknowledgments:** T.C. gratefully acknowledges support by the NSF through the grant DMS-2009800, and the RTG Grant DMS-1840314 - _Analysis of PDE_. P.M.E. was supported by NSF grant DMS-2009800 through T.C.
