Deep Learning – SGI 2025

SGI Fellows:
Stephanie Atherton, Marina Levay, Ualibyek Nurgulan, Shree Singhi, Erendiro Pedro

In the Topology Control project mentored by Professor Paul Kry and project assistants Daria Nogina and Yuanyuan Tao, we sought to explore preserving topological invariants of meshes within the framework of DeepSDFs. Deep Signed Distance Functions are a neural implicit representation used for shapes in geometry processing, but they don’t come with the promise of respecting topology. After finishing our ML pipeline, we explored various topology-preserving techniques through our simple, initial case of deforming a “donut” (torus) into a mug.

DeepSDFs

Signed Distance Field (SDF) representation of a 3D bunny. The network predicts the signed distance from each spatial point to the surface. Source: (Park et al., 2019).

Signed Distance Functions (SDFs) return the shortest distance from any point in 3D space to the surface of an object. Their sign indicates spatial relation: negative if the point lies inside, positive if outside. The surface itself is defined implicitly as the zero-level set: the locus where $\text{SDF}(x) = 0 $.

In 2019, Park et al. introduced DeepSDF, the first method to learn a continuous SDF directly using a deep neural network (Park et al., 2019). Given a shape-specific latent code $ z \in \mathbb{R}^d $ and a 3D point $ x \in \mathbb{R}^3 $, the network learns a continuous mapping:

$$
f_\theta(z_i, x) \approx \text{SDF}^i(x),
$$

where $ f_\theta $ takes a latent code $ z_i $ and a 3D query point $ x $ and returns an approximate signed distance.

The training set is defined as:

$$
X := {(x, s) : \text{SDF}(x) = s}.
$$

Training minimizes the clamped L1 loss between predicted and true distances:

$$
\mathcal{L}\bigl(f_\theta(x), s\bigr)
= \bigl|\text{clamp}\bigl(f_\theta(x), \delta\bigr) – \text{clamp}(s, \delta)\bigr|
$$

with the clamping function:

$$
\text{clamp}(x, \delta) = \min\bigl(\delta, \max(-\delta, x)\bigr).
$$

Clamping focuses the loss near the surface, where accuracy matters most. The parameter $ \delta $ sets the active range.

This is trained on a dataset of 3D point samples and corresponding signed distances. Each shape in the training set is assigned a unique latent vector $ z_i $, allowing the model to generalize across multiple shapes.

Once trained, the network defines an implicit surface through its decision boundary, precisely where $ f_\theta(z, x) = 0 $. This continuous representation allows smooth shape interpolation, high-resolution reconstruction, and editing directly in latent space.

Training Field Notes

We sampled training data from two meshes, torus.obj and mug.obj using a mix of blue-noise points near the surface and uniform samples within a unit cube. All shapes were volume-normalized to ensure consistent interpolation.

DeepSDF is designed to intentionally overfit. Validation is typically skipped. Effective training depends on a few factors: point sample density, network size, shape complexity, and sufficient epochs.

After training, the implicit surface can be extracted using Marching Cubes or Marching Tetrahedra to obtain a polygonal mesh from the zero-level set.

Training Parameters
SDF Delta	1.0
Latent Mean	0.0
Latent SD	0.01
Loss Function	Clamped L1
Optimizer	Adam
Network Learning Rate	0.001
Latent Learning Rate	0.01
Batch Size	2
Epochs	5000
Max Points per Shape	3000
Network Architecture
Latent Dimension	16
Hidden Layer Size	124
Number of Layers	8
Input Coordinate Dim	3
Dropout	0.0
Point Cloud Sampling
Radius	0.02
Sigma	0.02
Mu	0.0
Number of Gaussians	10
Uniform Samples	5000

For higher shape complexity, increasing the latent dimension or training duration improves reconstruction fidelity.

Latent Space Interpolation

One compelling application is interpolation in latent space. By linearly blending between two shape codes $ z_a $ and $ z_b $, we generate new shapes along the path

$$
z(t) = (1 – t) \cdot z_a + t \cdot z_b,\quad t \in [0,1].
$$

Latent space interpolation between mug and torus.

While DeepSDF enables smooth morphing between shapes, it exposes a core limitation: a lack of topological consistency. Even when the source and target shapes share the same number of genus, interpolated shapes can exhibit unintended holes, handles, or disconnected components. These are not artifacts, they reveal that the model has no built-in notion of topology.

However, this limitation also opens the door for deeper exploration. If neural fields like DeepSDF lack an inherent understanding of topology, how can we guide them toward preserving it? In the next post, we explore a fundamental topological property—genus—and how maintaining it during shape transitions could lead us toward more structurally meaningful interpolations.

References

https://arxiv.org/pdf/1901.05103

Quivers are a tool that are known to help us simplify problems in math. In particular, representations of quivers contribute to geometric perspectives in representation theory: the theory of reducing complex algebraic structures to simpler ones. Lesser known, neural networks can also be represented using quiver representation theory.

Fundamentally, a quiver is just a directed graph.

Intrinsic definitions to consider include:

A source vertex of a quiver has no edges directed towards it
A sink vertex has no edges directed away from it
A loop in a quiver is an oriented edge such that the start vertex is the same as the end vertex

A fancy type of quiver known as an Auslander-Reiten quiver, courtesy of the author. But remember!, a quiver is simply a directed graph.

Just like an MLP, a network quiver $Q$ is arranged by input, output, and hidden layers in between. Likewise, they also have input vertices (a subset of source vertices), bias vertices (the source vertices that are not input vertices), and output vertices (sinks of $Q$). All remaining vertices are hidden vertices. The hidden quiver $\tilde{Q}$ consists of all hidden vertices $\tilde{V}$ of $Q$ and all oriented edges $\tilde{E}$ between $\tilde{V}$ of $Q$ that are not loops.

Def: A network quiver $Q$ is a quiver arranged by layers such that:

There are no loops on source (input and bias) nor sink vertices.
There exists exactly one loop on each hidden vertex

For any quiver $Q$, we can also define its representation $\mathcal{Q}$, in which we assign a vector space to each vertex of $Q$ and regard our directed edges of $Q$ as $k$-linear maps. In a thin representation, each $k$-linear map is simply a $1\times1$ matrix.

A quiver with 4 vertices, courtesy of the author.

A *representation* of the quiver directly above, courtesy of the author.

Defining a neural network $(W, f)$ over a network quiver $Q$, where $W$ is a specific thin representation and $f = (f_v)_{v \in V}$ are activation functions, allows much of the language and ideas of quiver representation theory to carry over to neural networks .

When a neural network like an MLP does its forward pass, it gives rise to a pointwise activation function $f$, defined here as a one variable non-linear function $f: \mathbb{C} \to \mathbb{C}$ differentiable except in a set of measure zero. We assign these activation functions to loops of $Q$.

Further, for a neural network $(W, f)$ over $Q$, we have a network function

$$ \Psi(W, f): \mathbb{C}^d \to \mathbb{C}^k $$

where the coordinates of $\Psi(W, f)(x)$ are the score of the neural net as the activation outputs of the output vertices of $(W, f)$ with respect to an input data vector $x \in \mathbb{C}^d$.

The manifold hypothesis critical to deep learning proposes that high-dimensional data actually lies in a low-dimensional, latent manifold within the input space. We can map the input space to the geometric moduli space of neural networks $_d\mathcal{M}_k(\tilde{Q})$ so that our latent manifold is also translated to the moduli space. While $_d\mathcal{M}_k(\tilde{Q})$ depends on the combinatorial structure of the neural network, activation and weight architectures of the neural network determine how data is distributed inside the moduli space.

The mobius strip is a well-known geometric manifold.

We will approach the manifold hypothesis via framed quiver representations. A choice of a thin representation $\tilde{\mathcal{Q}}$ of the hidden quiver $\tilde{Q}$ and a map $h$ from the hidden representation $\tilde{\mathcal{Q}}$ to hidden vertices determine a pair $(\tilde{\mathcal{Q}}, h)$, where $h = \{h_v\}{v \in \tilde{V}}$. The pair $(\tilde{\mathcal{Q}}, h)$ is used to denote our framed quiver representation.

Def: A double-framed thin quiver representation is a triple $(l, \tilde{\mathcal{Q}}, h)$ where:

$\tilde{\mathcal{Q}}$ is a thin representation of the hidden quiver $\tilde{Q}$
$(\tilde{\mathcal{Q}}, h)$ is framed representation of $\tilde{Q}$
$(\tilde{\mathcal{Q}}, l)$ is a co-framed representation of $\tilde{Q}$ (the dual of a framed representation)

Denote by $_d\mathcal{R}_k(\tilde{\mathcal{Q}})$ the space of all double-framed thin quiver representations. We will use stable double-framed thin quiver representations in our construction of moduli space.

Def: A double-framed thin quiver representation $\texttt{W}_k^f = (l, \tilde{\mathcal{Q}}, h)$ is stable if :

The only sub-representation of $\tilde{\mathcal{Q}}$ contained in the kernel of $h$ is the zero sub-representation
The only sub-representation of $\tilde{\mathcal{Q}}$ contained in the image of $l$ is $\tilde{\mathcal{Q}}$

Def: We present the moduli space of double-framed thin quiver representations as

$$ _d\mathcal{M}_k(\tilde{Q}):=\{[V]: _d\mathcal{R}_k(\tilde{\mathcal{Q}}) \space \text{is stable} \}. $$

The moduli space depends on the hidden quiver as well as the chosen vector spaces. Returning to neural networks $(W, f)$, and given an input data vector $x \in \mathbb{C}^d$, we can define a map

$$ \varphi(W, f): \mathbb{C}^d \to _d\mathcal{R}_k(\tilde{\mathcal{Q}})\\x \mapsto \texttt{W}_k^f. $$

This map takes values in the moduli space, the points of which parametrize isomorphism classes of stable double-framed thin quiver representations. Thus we have

$$ \varphi(W, f): \mathbb{C}^d \to _d\mathcal{M}_k(\tilde{Q}).
$$

As promised, we have mapped our input space containing our latent manifold to the moduli space $_d\mathcal{M}_k(\tilde{Q})$ of neural networks, mathematically validating the manifold hypothesis.

Independent of the architecture, activation function, data, or task, any decision of any neural network passes through the moduli (as well as representation) space. With our latent manifold translated into the moduli space, we have an algebro-geometric way to continue to study the dynamics of neural network training.

Looking through the unsuspecting the lens of quiver representation theory has the potential to provide new insights in deep learning, where network quivers appear as a combinatorial tool for understanding neural networks and their moduli spaces. More concretely:

Continuity and differentiability of the network function $\Psi(W, f)$ and map $\varphi(W, f)$ should allow us to apply further algebro-geometric tools to the study of neural networks, including to our constructed moduli space $_d\mathcal{M}_k(\tilde{Q})$.
Hidden quivers can aid us in comprehending optimization hyperparameters in deep learning. We may be able to transfer gradient descent optimization to the setting of the moduli space.
Studying training within moduli spaces can lead to the development of new convergence theorems to guide deep learning.
The dimension of $_d\mathcal{M}_k(\tilde{Q})$ could be used to quantify the capacity of neural networks.

The manifold hypothesis has played a ubiquitous role throughout deep learning since originally posed, and formalizing its existence via the moduli of quiver representations can help us to understand and potentially improve upon the effectiveness of neural networks and their latent spaces.

Notes and Acknowledgements. Content for this post was largely borrowed from and inspired by The Representation Theory of Neural Networks, smoothing over many details more rigorously presented in the original paper. We thank the 2025 SGI organizers and sponsors for supporting the author’s first deep learning-related research experience via the “Topology Control” project as well as mentors and other research fellows involved for their diverse expertise and patience.

Author

stephanie.p.atherton