arxiv: 2604.26942 · v1 · submitted 2026-04-29 · 💻 cs.LG · math.ST· q-bio.GN· stat.ME· stat.ML· stat.TH

Recognition: unknown

Hyper Input Convex Neural Networks for Shape Constrained Learning and Optimal Transport

Insung Kong, Johannes Schmidt-Hieber, Shayan Hundrieser

Authors on Pith no claims yet

Pith reviewed 2026-05-07 08:21 UTC · model grok-4.3

classification 💻 cs.LG math.STq-bio.GNstat.MEstat.MLstat.TH

keywords input convex neural networksmaxout networksoptimal transportconvex regressionshape constrained learningneural optimal transporthigh-dimensional approximationsingle-cell data

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The pith

HyCNNs require exponentially fewer parameters than ICNNs to approximate quadratic functions to a given precision.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces Hyper Input Convex Neural Networks (HyCNNs) that combine maxout units with input convex neural network designs to guarantee convexity in the input while allowing depth to improve expressivity. The central proof shows that this structure approximates any quadratic function to fixed precision using exponentially fewer parameters than standard ICNNs. Experiments on synthetic convex regression and interpolation tasks demonstrate lower error rates than both ICNNs and ordinary MLPs. The same networks are applied to learning high-dimensional optimal transport maps, where they match or exceed ICNN-based methods on synthetic distributions and single-cell RNA data.

Core claim

HyCNNs integrate maxout activations into the ICNN framework so that the output remains convex in the input by construction while depth can be used effectively. The key theoretical result is that the parameter count needed to approximate quadratic functions to any given accuracy scales exponentially better than in prior ICNN designs. This efficiency translates into more stable training at scale and stronger performance on convex regression, interpolation, and optimal transport map estimation tasks.

What carries the argument

Hyper Input Convex Neural Networks (HyCNNs), which replace selected layers in ICNNs with maxout units to preserve guaranteed input convexity while improving parameter efficiency and depth utilization.

If this is right

HyCNNs achieve any fixed approximation error on quadratic targets with exponentially fewer parameters than ICNNs.
HyCNNs produce lower prediction error than ICNNs and standard MLPs on synthetic convex regression and interpolation tasks.
HyCNNs learn high-dimensional optimal transport maps that often outperform those obtained from ICNN-based neural optimal transport methods on both synthetic and single-cell RNA datasets.
HyCNN training remains reliable when the network depth and width are increased, addressing a known limitation of earlier ICNN constructions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The exponential parameter savings could enable convex-constrained models in problem dimensions where ICNNs become computationally infeasible.
The architecture may transfer to other tasks that require monotonicity or convexity guarantees, such as utility function estimation or certain physics-informed learning problems.
Practitioners could adopt HyCNNs as a drop-in replacement in existing optimal transport pipelines to gain accuracy without altering the overall optimization setup.

Load-bearing premise

The approach assumes that the target functions of interest are well approximated by the HyCNN structure and that input convexity together with training stability continue to hold when the networks are scaled up.

What would settle it

Compare the smallest number of parameters required by a HyCNN versus a standard ICNN to approximate the squared Euclidean norm (a simple quadratic) to within a fixed small error in dimension 10 or higher; absence of an exponential gap in parameter count would refute the efficiency claim.

Figures

Figures reproduced from arXiv: 2604.26942 by Insung Kong, Johannes Schmidt-Hieber, Shayan Hundrieser.

**Figure 1.** Figure 1: Input convex neural networks (ICNNs) versus Hyper Input Convex Neural Networks (HyCNNs). (a) Depiction of generic input convex network architecture. ICNN blocks comprise one lane, whereas Hyper ICNN (HyCNN) blocks involve two lanes of matrix operations. (b) Averaged Empirical MSE over 10 repetitions with (10% − 90%) confidence bands for learning the function f0(x) = ∥x∥ 2 2 on R d with d = 50 based on n = … view at source ↗

**Figure 2.** Figure 2: Signal propagation and gradient profiles across relative depth in HyCNN at initialization for width W = 48 and depths L ∈ {2, 4, . . . , 16} (colors) for input dimension d = 50. Relative depth denotes the respective layer index normalized by network depth. (a) Signals are computed from a single forward pass at initialization with standard Gaussian input X ∼ N (0, Id). Each curve is the mean hidden-state ℓ2… view at source ↗

**Figure 3.** Figure 3: Pushforwards under learned HyCNN OT map (width 48, depth 4). The HyCNN is trained for n = m = M = 2000 data points for 2500 outer iterations for learning f, with S = 10 steps per iterations for learning g, via Adam and decaying learning rates and smoothness parameter τ . 5 Experiments 5.1 Convex regression We draw covariates (Xi) n i=1 from Unif[−1, 1] d with responses according to the model Yi = f(Xi) + ϵ… view at source ↗

**Figure 4.** Figure 4: Prediction MSE for different training sample sizes n for the 50-dimensional regression tasks with 10th-90th percentile bands across 10 runs. Each panel shows HyCNNs with depths 2, 4, 8, 16 and the best MLP, ICNN, GMN at n = 104 among depths ∈ {2, 4, 8, 16}. 10 view at source ↗

**Figure 5.** Figure 5: Performance of OT map estimation for synthetic and 4i single cell RNA-seq data. (a). Prediction MSE (normalized by dimension d = 50) for n = 5000 training samples for OT map estimation tasks with 10th-90th percentile bands across 10 runs. Both panels show the same HyCNN and the best ICNN with ReLU or leaky ReLU, or with a quadratic term in the first layer with ReLU or Softplus, and the Monge Gap MLP estima… view at source ↗

**Figure 6.** Figure 6: Illustration of the proof of Theorem 3.3 for L = 2 and (d1, d2) = (4, 4). uℓ(x), φℓ(x) and ψℓ(x) are defined by (B.4), (B.5) and (B.6), respectively. From (B.7) and (B.8), z2,0(x) = u1(x) and z2,j (x) = max(φ1(x), ψ1(x) + 7 16x − 7j 256 ) for j ∈ {1, 2, 3}. Gray vertical lines are drawn at the kink locations of the functions. 22 view at source ↗

**Figure 7.** Figure 7: Illustration of the proof of Proposition A.2 for d = 8 and p = q = 3. Proof of Proposition A.2. We write x = (x1, . . . , xd) ⊺ . We define κ ∶ N → 2N0 as κ(n) ∶= ⎧⎪⎪ ⎨ ⎪⎪⎩ n n is even, n − 1 n is odd. Consider arbitrary p, q ∈ N with pq ≥ d. By Theorem 3.3, there exists a HyCNN g∶ R → R with ⌊L/p⌋ hidden layers, κ(⌊(m − 1)/q⌋) neurons per layer and σ(a, b) = max(a, b), such that sup x∈[0,1] ∣g(x) − x 2 ∣ … view at source ↗

**Figure 8.** Figure 8: HyCNNs at initialization across different depths. Each panel shows a single randomly initialized HyCNN according to the initialization scheme from Section 2.2 with maxout activation, input dimension d = 2, domain x ∈ [−10, 10] 2 , width m = 32, and depths L ∈ {2, 4, 6, 8}. The surface plot is shown together with its contour projection on the base plane. The visualization demonstrates that the initialized f… view at source ↗

**Figure 9.** Figure 9: Univariate regression results (d = 1) for the six ground-truth functions (a–f) from Section D.2. Each panel shows the fitted curves of HyCNNs (width 48) with depths L ∈ {2, 4, 8, 16, 32} and the best MLP (width 64), ICNN (width 64), or GMN (width 48) among the same depths; the green dots depict the training samples and the black curve depicts the ground-truth function. For each function, the visualization … view at source ↗

**Figure 10.** Figure 10: Prediction MSE for different training sample sizes n for the 50-dimensional regression tasks with 10th-90th percentile bands across 10 runs. Each panel shows HyCNNs with depths 2, 4, 8, 16 and the best MLP, ICNN, GMN at n = 104 among similar depths. Across all settings, the key observations are consistent: HyCNNs with depth at least four consistently achieve the lowest prediction MSE, particularly in high… view at source ↗

**Figure 11.** Figure 11: Reverse OT map estimation in dimension d = 2. Each panel shows the pushforward of the target distribution under the learned reverse HyCNN OT map (width 48, depth 4), trained with n = m = M = 2000 data points, T = 2500 outer iterations, and S = 5 inner steps, using Adam with cosine-decayed learning rate (λ = 10−2 , final ratio 0.01) and smoothness parameter τ0 = 1 (cosine decay). From left to right: (i) fi… view at source ↗

**Figure 12.** Figure 12: Contour plots of the learned OT potential φ̂ in dimension d = 2. Each panel shows the contour lines of the HyCNN potential (width 48, depth 4) learned from n = m = M = 2000 data points for the same collection of source–target pairs as in view at source ↗

**Figure 13.** Figure 13: Enlarged view of view at source ↗

read the original abstract

We introduce Hyper Input Convex Neural Networks (HyCNNs), a novel neural network architecture designed for learning convex functions. HyCNNs combine the principles of Maxout networks with input convex neural networks (ICNNs) to create a neural network that is always convex in the input, theoretically capable of leveraging depth, and performs reliable when trained at scale compared to ICNNs. Concretely, we prove that HyCNNs require exponentially fewer parameters than ICNNs to approximate quadratic functions up to a given precision. Throughout a series of synthetic experiments, we demonstrate that HyCNNs outperform existing ICNNs and MLPs in terms of predictive performance for convex regression and interpolation tasks. We further apply HyCNNs to learn high-dimensional optimal transport maps for synthetic examples and for single-cell RNA sequencing data, where they oftentimes outperform ICNN-based neural optimal transport methods and other baselines across a wide range of settings.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

HyCNNs mix maxout into ICNNs for a claimed exponential parameter cut on quadratics plus some OT gains, but the proof's baseline choice needs checking before the efficiency story lands.

read the letter

HyCNNs are input-convex networks that add maxout units under the usual non-negative weight constraints. The central new claim is a proof that this version approximates quadratic functions to precision epsilon with exponentially fewer parameters than a standard ICNN. They also report better predictive performance than ICNNs and MLPs on synthetic convex regression and interpolation, and they apply the model to high-dimensional optimal transport on both synthetic data and single-cell RNA sequencing, where it beats ICNN-based transport methods in several settings. The construction keeps convexity by design and appears easier to train at scale than plain ICNNs. That combination is the main practical takeaway. The experiments give a reasonable first check on the architecture in the OT setting, which is a natural use case for convex potentials. The soft spot sits in the theory. The exponential gap is shown against a basic ICNN with ReLU-style activations and fixed positive weights. If a comparable maxout construction can be folded into the standard ICNN framework without losing convexity or reintroducing the same parameter blow-up, the advantage shrinks. The paper should clarify whether the baseline already incorporates the strongest possible ICNN variant or whether the comparison is to a deliberately limited one. Experimental details on variance, exact hyperparameter matching, and scaling behavior are also thin in the high-level summary, so the reported outperformance could be narrower once those are filled in. This work is aimed at researchers who build neural architectures for convex optimization or optimal transport. Readers who care about parameter-efficient convex models will find the construction and the OT experiments useful. It is solid enough on its own terms to deserve a serious referee, mainly to pressure-test the proof against stronger baselines and to verify the empirical protocols. I would send it to review rather than desk-reject.

Referee Report

3 major / 3 minor

Summary. The paper introduces Hyper Input Convex Neural Networks (HyCNNs) that combine maxout networks with input convex neural networks (ICNNs) to produce models that are convex in the input. The central claim is a proof that HyCNNs require exponentially fewer parameters than ICNNs to approximate quadratic functions to a given precision. The authors further report that HyCNNs outperform ICNNs and MLPs on synthetic convex regression and interpolation tasks and yield competitive or superior results when used to learn high-dimensional optimal transport maps on both synthetic data and single-cell RNA sequencing data.

Significance. If the exponential parameter-reduction result survives comparison to strengthened ICNN baselines and the reported empirical gains are statistically reliable, the architecture could improve scalability for shape-constrained regression and neural optimal transport. The explicit use of maxout units to increase expressivity while preserving convexity is a concrete technical contribution, and the real-data OT experiments provide a useful existence proof of practical applicability.

major comments (3)

[§4, Theorem on quadratic approximation] §4 (theoretical analysis, Theorem on quadratic approximation): The proof that HyCNNs require exponentially fewer parameters than ICNNs for quadratic approximation compares against a standard ICNN using ReLU-style activations and fixed positive weights. It does not address whether an ICNN variant that incorporates maxout units while still obeying the non-negative input-weight constraints required for convexity could achieve comparable scaling. Without ruling out or comparing against such variants, the claimed exponential gap may be an artifact of the chosen baseline rather than an intrinsic advantage of the HyCNN construction.
[§5] §5 (experimental evaluation): The synthetic and real-data results claim consistent outperformance, yet the manuscript provides no error bars, standard deviations across random seeds, or detailed descriptions of hyperparameter selection and network-depth choices. For the single-cell RNA-seq optimal-transport experiments, it is unclear how the high-dimensional maps were regularized and whether post-training convexity was verified numerically.
[§3] §3 (architecture definition): The precise weight constraints and activation rules that guarantee input-convexity after the maxout combination are stated at a high level but lack an explicit inductive proof or set of sufficient conditions that survive depth scaling. This detail is load-bearing for the claim that HyCNNs remain convex while leveraging depth.

minor comments (3)

[Abstract] Abstract: 'performs reliable when trained at scale' should read 'performs reliably when trained at scale'.
[§5] Figures in §5: Add captions that explicitly state the plotted metric (e.g., mean squared error, Wasserstein distance) and whether shaded regions represent standard error over multiple runs.
[§3] Notation: The definition of the hyper-network parameters and how they interact with the maxout units should be introduced with a single consolidated equation block rather than scattered across paragraphs.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their detailed and constructive feedback on our manuscript. We address each of the major comments below and outline the revisions we will make to strengthen the paper.

read point-by-point responses

Referee: [§4, Theorem on quadratic approximation] The proof that HyCNNs require exponentially fewer parameters than ICNNs for quadratic approximation compares against a standard ICNN using ReLU-style activations and fixed positive weights. It does not address whether an ICNN variant that incorporates maxout units while still obeying the non-negative input-weight constraints required for convexity could achieve comparable scaling. Without ruling out or comparing against such variants, the claimed exponential gap may be an artifact of the chosen baseline rather than an intrinsic advantage of the HyCNN construction.

Authors: We appreciate this insightful comment. Our theoretical analysis in §4 establishes the parameter efficiency of HyCNNs relative to the standard ICNN architecture as introduced in prior work. The HyCNN construction integrates maxout units in a manner that preserves input convexity through a hypernetwork parameterization, which differs from simply augmenting an ICNN with maxout while enforcing non-negative weights on input connections. We acknowledge that a direct comparison to a hypothetical maxout-enhanced ICNN variant would further strengthen the result. In the revised manuscript, we will add a discussion in §4 clarifying the distinction between our approach and such variants, and note that no such maxout-ICNN has been proposed or analyzed in the literature to date. revision: partial
Referee: [§5] The synthetic and real-data results claim consistent outperformance, yet the manuscript provides no error bars, standard deviations across random seeds, or detailed descriptions of hyperparameter selection and network-depth choices. For the single-cell RNA-seq optimal-transport experiments, it is unclear how the high-dimensional maps were regularized and whether post-training convexity was verified numerically.

Authors: Thank you for highlighting these important aspects of the experimental section. We agree that the current presentation lacks sufficient statistical rigor and implementation details. In the revised version, we will include error bars and standard deviations computed over multiple random seeds for all synthetic and real-data experiments. We will also expand the experimental setup subsection to provide full details on hyperparameter selection, network architectures, and depth choices. For the single-cell RNA-seq OT experiments, we will describe the regularization techniques employed and report numerical verification of post-training convexity, such as checking the Hessian or gradient monotonicity on held-out samples. These changes will be incorporated as a full revision to §5. revision: yes
Referee: [§3] The precise weight constraints and activation rules that guarantee input-convexity after the maxout combination are stated at a high level but lack an explicit inductive proof or set of sufficient conditions that survive depth scaling. This detail is load-bearing for the claim that HyCNNs remain convex while leveraging depth.

Authors: We thank the referee for pointing out the need for greater rigor in the architectural definition. While §3 outlines the weight constraints and the use of maxout to maintain convexity, we agree that an explicit inductive proof would enhance clarity. In the revised manuscript, we will include a formal inductive proof in §3 or an appendix demonstrating that the HyCNN architecture preserves input convexity at arbitrary depth under the specified constraints. This will involve showing that each layer's output remains convex in the input when composed appropriately. This constitutes a full revision to address the concern. revision: yes

Circularity Check

0 steps flagged

No circularity: HyCNN parameter-efficiency claim is a new construction with independent proof content

full rationale

The paper defines HyCNNs as a novel combination of maxout units with the non-negative weight constraints of ICNNs, then states a separate theorem proving exponential parameter reduction for quadratic approximation. No quoted step reduces the claimed advantage to a fitted parameter renamed as prediction, a self-citation chain, or an ansatz smuggled from prior work by the same authors. The baseline ICNN is the standard construction from external literature; the proof is presented as comparing against that fixed baseline rather than re-deriving it from HyCNN itself. Empirical sections on regression and OT maps are downstream validations, not load-bearing for the theoretical claim. The derivation chain therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central contribution is the new HyCNN architecture itself. No free parameters are mentioned. The main domain assumption is that the functions to be learned are convex. The invented entity is the HyCNN itself.

axioms (1)

domain assumption Target functions are convex
The entire architecture and all experiments are built around enforcing and learning convex functions.

invented entities (1)

Hyper Input Convex Neural Network (HyCNN) no independent evidence
purpose: A neural network that is always convex in the input, leverages depth, and approximates quadratics with exponentially fewer parameters than ICNNs
New architecture introduced and analyzed in the paper; no independent evidence outside this work is provided in the abstract.

pith-pipeline@v0.9.0 · 5474 in / 1253 out tokens · 57199 ms · 2026-05-07T08:21:08.675048+00:00 · methodology

discussion (0)

Reference graph

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