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Polyhedral Complex Derivation from Piecewise Trilinear Networks

Jin-Hwa Kim

TL;DR

Focusing on trilinear interpolating methods as positional encoding, theoretical insights are presented and an analytical mesh extraction is presented, showing the transformation of hypersurfaces to flat planes within the trilinear region under the eikonal constraint.

Abstract

Recent advancements in visualizing deep neural networks provide insights into their structures and mesh extraction from Continuous Piecewise Affine (CPWA) functions. Meanwhile, developments in neural surface representation learning incorporate non-linear positional encoding, addressing issues like spectral bias; however, this poses challenges in applying mesh extraction techniques based on CPWA functions. Focusing on trilinear interpolating methods as positional encoding, we present theoretical insights and an analytical mesh extraction, showing the transformation of hypersurfaces to flat planes within the trilinear region under the eikonal constraint. Moreover, we introduce a method for approximating intersecting points among three hypersurfaces contributing to broader applications. We empirically validate correctness and parsimony through chamfer distance and efficiency, and angular distance, while examining the correlation between the eikonal loss and the planarity of the hypersurfaces.

Polyhedral Complex Derivation from Piecewise Trilinear Networks

TL;DR

Focusing on trilinear interpolating methods as positional encoding, theoretical insights are presented and an analytical mesh extraction is presented, showing the transformation of hypersurfaces to flat planes within the trilinear region under the eikonal constraint.

Abstract

Recent advancements in visualizing deep neural networks provide insights into their structures and mesh extraction from Continuous Piecewise Affine (CPWA) functions. Meanwhile, developments in neural surface representation learning incorporate non-linear positional encoding, addressing issues like spectral bias; however, this poses challenges in applying mesh extraction techniques based on CPWA functions. Focusing on trilinear interpolating methods as positional encoding, we present theoretical insights and an analytical mesh extraction, showing the transformation of hypersurfaces to flat planes within the trilinear region under the eikonal constraint. Moreover, we introduce a method for approximating intersecting points among three hypersurfaces contributing to broader applications. We empirically validate correctness and parsimony through chamfer distance and efficiency, and angular distance, while examining the correlation between the eikonal loss and the planarity of the hypersurfaces.
Paper Structure (45 sections, 15 theorems, 50 equations, 14 figures, 10 tables, 4 algorithms)

This paper contains 45 sections, 15 theorems, 50 equations, 14 figures, 10 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Related work
  3. Mesh extraction in deep neural networks.
  4. Positional encoding.
  5. Eikonal equation and SDF.
  6. Preliminaries
  7. Tropical geometry
  8. Edge subdivision for tropical geometry
  9. Method
  10. Piecewise trilinear networks
  11. Curved edge subdivision in a trilinear space
  12. Skeletonization, faces, and normals
  13. Implementation
  14. Initialization.
  15. Optimizing sign-vectors.
  16. ...and 30 more sections

Key Result

Proposition 3.3

The decision boundary for $L$-layer neural networks with ReLU activation $\nu^{(L)}({\bm{x}}) = 0$ is a subset of or equals with the region where two tropical monomials, or two arguments of $\max$, equal (def:poly), as follows: where the subscript $j$ denotes the $j$-th neuron, assuming the hidden size of neural networks is $H$, while the final $L$-th layer has a single output for rather discussi

Figures (14)

  • Figure 1: The mesh is analytically extracted from piecewise trilinear networks, comprising both HashGrid muller2022instant and ReLU neural networks, that have been trained to learn a signed distance function with the eikonal loss. We start with initial vertices and edges defined using the grid marks (1), and its linear subdivision (2a); however, intersecting polygons (ref.\ref{['sec:edge_subdivision']}) need bilinear subdivision (2b) and consequentially trilinear subdivision (2c). Note that the linear and bilinear subdivisions are specific instances of trilinear subdivisions with straightforward solutions.
  • Figure 2: Trilinear regions in the xy-plane at $z = 0.04$, identified by the sign-vectors (\ref{['def:signv']}), are represented with random colors. (a) Grids described in \ref{['sec:implementation']} and \ref{['alg:hashgrid_marks']}. (b) The neurons of the first layer representing folded hypersurfaces (blue arrow). (c) All neurons representing every nonlinear boundary. (d) Select all zero-set vertices and edges. (e) Skeletonized as in \ref{['sec:skeletonization']}.
  • Figure 3: Chamfer distance for the bunny with the Large model comparing with MC, MT, and NDC.
  • Figure 4: Chamfer distances for the Stanford dragon varying the number of vertices and the model sizes, showing consistent efficiency.
  • Figure 5: Effect of the weight of eikonal loss on the flatness error in \ref{['eqn:flatness']}. For the plot, we conduct experiments using the unit sphere.
  • ...and 9 more figures

Theorems & Definitions (32)

  • Definition 3.1: Tropical hypersurface
  • Definition 3.2: ReLU neural networks
  • Proposition 3.3: Decision boundary of neural networks
  • Definition 3.4: Sign-vectors
  • Definition 4.1: Trilinear interpolation
  • Lemma 4.2: Nested trilinear interpolation
  • proof
  • Definition 4.3: Piecewise trilinear networks
  • Lemma 4.4: Curved edge of two hypersurfaces
  • proof
  • ...and 22 more