Brownian Heads for Deep ReLU Representations: Activation Mass and the Cost of Same-Sample Selection

We study Gaussian regression over the explicit vector-valued Parhi--Nowak deep-RBV^2 architecture with depth L, width w, layer-sum variation budget A, and output bound B. For this O(L w^2)-parameterized architecture, the known lower and upper bounds differ by one factor of depth. We construct a local packing showing that the quadratic depth dependence is intrinsic under an explicit sample-size-dependent radius condition. The packing has log-cardinality Omega(L^2 w^2 log w); its codewords lie in an O(lambda) L^2 ball and are pairwise Omega(lambda)-separated. The main ingredients are a bias-corrected bounded-coefficient approximation theorem and balanced amplification: multiplying a depth-D ReLU network by q can be implemented using one constant channel so that every coefficient grows by only q^(1/D). Translation to vector-valued RBV^2 blocks then has layer-sum cost O(D w^2 q^(1/D)). Gaussian Fano yields a radius-explicit lower bound governed by the output, testing, and representation scales. Under A=B=R, sigma proportional to R, and the stated radius condition, this gives minimax risk at least of order L^2 w^2 log(w) R^2/n. A pseudodimension-based finite-net upper bound gives O-tilde(L^2 w^2 R^2/n) for unbounded Gaussian responses. Thus the minimax risk has quadratic polynomial dependence on depth, up to logarithmic factors, and exhibits a transition to representation-limited behavior at smaller radius.

We develop a unified function space theory of deep fully connected neural networks. Functions in our spaces are defined recursively as $\ell^1$-bounded linear combinations of activated functions from preceding layers, with a dictionary of affine functions at the first layer. Unlike existing theories that are largely specialized to homogeneous activations such as the ReLU, our framework provides a meaningful notion of functional complexity for deep networks with a broad range of homogeneous and non-homogeneous activation functions commonly used in practice. This simple construction unites several seemingly disparate ideas from the literature, including norm-based complexity bounds and variational characterizations of depth, and facilitates novel analyses of what kinds of functions deep norm-constrained networks can represent. To this end, we prove a novel representer theorem for our spaces and establish novel function-space complexity bounds showing that the associated function classes remain qualitatively small at arbitrary depth. In the univariate ReLU case, we prove a "depth saturation" result: depth in this setting yields only a small constant rescaling of the function class, with no added functional diversity. As a consequence, we show that deep norm-controlled ReLU functions in any dimension cannot exhibit high frequencies along any direction. This finding reveals that some commonly cited expressivity benefits of depth disappear once network complexity is controlled by an appropriate function space norm, rather than parameter count or other representational costs that permit compounded rescaling across layers. Overall, our results illustrate how a function space perspective yields new structural insights into the relationship between depth and complexity.

An input may activate few hidden units even when different inputs collectively use an entire network. We study the statistical complexity of this input-dependent sparsity in the one-hidden-layer ReLU model of Awasthi et al. (COLT 2024). For width $s$, at most $k$ active units per input, and effective weight and bias bounds $W,B$, every size-$m$ sample in the class's fixed radius-$R$ input domain satisfies $\mathcal{R}(S)\le CWR\min\{k,\sqrt{sk/m}\log^{3/2}(2m)\}+kB/\sqrt m$. A support-preserving cover and a single normalized chaining argument remove the previous explicit dimension factor, up to logarithms. Lower bounds on appropriate i.i.d. marginals match up to those logarithms, showing how changing active units across inputs retains a width dependence. The input domain matters: zero-bias networks sparse on the entire ball have at most $2k$ nonzero units and complexity $O(kWR/\sqrt m)$, whereas bias bounds comparable to $WR$ restore the worst-case rate on that same domain in only logarithmic dimension. A spherical-cap construction proves the latter claim without assuming sparsity merely on the sampling support. For a specified normalized bounded loss and biases comparable to $WR$, we also obtain agnostic minimax excess-risk bounds of order $\min\{1,\sqrt{s/(km)}\}$ up to logarithms.