Level-Crossing Density as a Mesh-Free High-Frequency Auxiliary Loss for Implicit Neural Representations

• URL: http://arxiv.org/abs/2607.05815v1
• Date: Tue, 07 Jul 2026 04:23:23 GMT
• Title: Level-Crossing Density as a Mesh-Free High-Frequency Auxiliary Loss for Implicit Neural Representations
• Abstract summary: We derive smooth Monte-Carlo estimators for a continuous neural field using no grid, no complex, no persistence.<n>The estimator is accurate to 1-3% against exact topology in 2D and 3D, and costs about 3 ms per iteration.<n>In 3D neural-SDF fitting, however, a failure mode we believe general to any sampled soft topology objective appears.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: The Minkowski functionals of a field's excursion sets -- area, boundary measure, and Euler characteristic -- describe its level-set morphology; the Euler characteristic is the cheapest handle on topology. We derive smooth Monte-Carlo estimators for all three of a continuous neural field, evaluated at scattered points via the co-area formula and Gauss-Bonnet, using only autodiff: no grid, no complex, no persistence. The estimator is accurate to 1-3% against exact topology in 2D and 3D, and costs about 3 ms per iteration where a persistent-homology (PH) loss on a cubical grid costs 650-1000 ms -- a 250x gap. We establish four design rules without which these losses silently fail: a dense level ladder (invariants are flat in the parameters away from transitions), a $C^2$ backbone (ReLU nets hide curvature in kinks), the full Minkowski vector (Euler characteristic alone is an alternating sum, gamed by debris-hole cancellation; pricing perimeter closes the channel), and sampling-scale coverage. In 2D the vector-valued cap is the only method in a controlled comparison that both repairs topology (3/3 seeds) and preserves fidelity -- uniform smoothing repairs at 11-17x the fidelity cost, and the Euler term alone repairs nothing. In 3D neural-SDF fitting, however, a failure mode we believe general to any sampled soft topology objective appears: gradient descent adversarially hides topological noise below the sampling density, where the estimator is blind -- spurious-feature counts are invariant to 4x more samples, and closing the window needs cubically many points, erasing the cost advantage. A grid-based PH baseline, whose complex is the evaluation resolution, solves the same benchmark ($4/9$ exact; median $b_1$ error 1 vs. ours above $10^4$). The 250x cost of persistence is, at present, the price of having no null space. We release estimators, receipts, and benchmarks.

Related papers

• Quantum Speedups for Stochastic Optimization with Heavy-Tailed Noise [49.730496294398726]
  We develop a novel quantum mean estimator for heavy-tailed random variables.<n>We derive stronger lower bounds for tail index>4/3$, showing that a nontrivial dependence on a dimension is unavoidable.<n>For a convex objective function, we propose a quantum projected gradient descent method.
  arXiv  Detail & Related papers  (2026-07-28T09:29:37Z)
• Efficient Mean Curvature Computation on High-Dimensional Data Manifolds [52.452902154360565]
  Estimating local mean curvature at each point of a high-dimensional dataset is a key ingredient of machine learning algorithms.<n>This paper introduces two complementary contributions that together reduce this cost by several orders of magnitude.<n> Experiments on real-world datasets confirm speedups of 50 to 300 times relative to the original implementation.
  arXiv  Detail & Related papers  (2026-06-04T16:04:31Z)
• Hidden-State Privacy Has an Empty Middle [51.56484100374058]
  We prove a complementary Fisher-ball lower bound: every full-rank Gaussian release at $O(1)$ Fisher utility admits a direction whose Mahalanobis signal grows linearly in hidden width.<n>A split-memory transformer trained from scratch reaches $G_mathrmMah in [20, 33]$ at 90M and maintains a $6$--$24times$ advantage over same-budget GPT baselines from 30M to 1B at a fixed-token language loss penalty.
  arXiv  Detail & Related papers  (2026-05-21T20:12:09Z)
• Intrinsic Wasserstein Rates for Score-Based Generative Models on Smooth Manifolds [61.14405512940818]
  We show that Score-based generative models are trained in high-dimensional ambient spaces.<n>We use a ReLU implementation of nearest-projection coordinates via finite intrinsic anchors and Gauss Newton.
  arXiv  Detail & Related papers  (2026-05-15T10:20:05Z)
• FLUIDSPLAT: Reconstructing Physical Fields from Sparse Sensors via Gaussian Primitives [31.072122193693854]
  We introduce FLUIDSPLAT, a sensor-conditioned model that predicts K anisotropic Gaussian primitives forming a partition-of-unity scaffold.<n>On a standard cylinder-flow benchmark, FLUIDSPLAT achieves the best mean error across all surface-sensor layouts.
  arXiv  Detail & Related papers  (2026-05-15T09:10:02Z)
• Wavelet Variance Equipartition as a Threshold for World-Model Quality and Quantum Kernel TN-Simulability [0.0]
  We identify the wavelet scaling exponent $$ as a critical diagnostic.<n>We establish $= 1/2$ as a sharp transition boundary for the classical simulability of amplitude-encoded quantum kernels.<n>We prove this variance scales strictly as $Var[X] = (d-2)$.
  arXiv  Detail & Related papers  (2026-05-12T05:41:12Z)
• Total Generalized Variation regularization closes the gap between neural-eld and classical methods in seismic travel-time tomography [0.0]
  Traveltime tomography forces a tradeoff between mesh resolution and stability in which the regularizer choice dominates what can be recovered.<n>We introduce MIMIR, a differentiable framework that represents the 2D velocity field as a Fourier-feature neural network.
  arXiv  Detail & Related papers  (2026-05-11T04:13:52Z)
• Problems with Chinchilla Approach 2: Systematic Biases in IsoFLOP Parabola Fits [35.1311414689253]
  Chinchilla Approach 2 biases imply a parameter underallocation corresponding to 6.5% of the $3.8times1025$ FLOP training budget.<n>Chinchilla Approach 3 largely eliminates these biases but is often regarded as less data-efficient, numerically unstable, and prone to local minima.<n>It may serve as a more convenient replacement for Approach 2 or a more scalable alternative for adaptations of Approach 3 to richer scaling law formulations.
  arXiv  Detail & Related papers  (2026-03-21T09:22:50Z)
• Monocular 3D Lane Detection via Structure Uncertainty-Aware Network with Curve-Point Queries [17.532686042065702]
  MonoUnc is a bird's-eye view (BEV)-free 3D lane detector that explicitly models aleatoric uncertainty informed by local lane structures.<n> Experiments on the ONCE-3DLanes and OpenLane datasets demonstrate that MonoUnc outperforms previous state-of-the-art (SoTA) methods.
  arXiv  Detail & Related papers  (2025-11-17T07:01:10Z)
• Learning with Norm Constrained, Over-parameterized, Two-layer Neural Networks [54.177130905659155]
  Recent studies show that a reproducing kernel Hilbert space (RKHS) is not a suitable space to model functions by neural networks. In this paper, we study a suitable function space for over- parameterized two-layer neural networks with bounded norms.
  arXiv  Detail & Related papers  (2024-04-29T15:04:07Z)
• Neural Inference of Gaussian Processes for Time Series Data of Quasars [72.79083473275742]
  We introduce a new model that enables it to describe quasar spectra completely. We also introduce a new method of inference of Gaussian process parameters, which we call $textitNeural Inference$. The combination of both the CDRW model and Neural Inference significantly outperforms the baseline DRW and MLE.
  arXiv  Detail & Related papers  (2022-11-17T13:01:26Z)
• Optimal Robust Linear Regression in Nearly Linear Time [97.11565882347772]
  We study the problem of high-dimensional robust linear regression where a learner is given access to $n$ samples from the generative model $Y = langle X,w* rangle + epsilon$ We propose estimators for this problem under two settings: (i) $X$ is L4-L2 hypercontractive, $mathbbE [XXtop]$ has bounded condition number and $epsilon$ has bounded variance and (ii) $X$ is sub-Gaussian with identity second moment and $epsilon$ is
  arXiv  Detail & Related papers  (2020-07-16T06:44:44Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.