Author: stephanie.p.atherton

Topology Control: Training a DeepSDF (1/2)

Post author By stephanie.p.atherton
Post date August 5, 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

Research

From Calculus to $k$-forms

Post author By stephanie.p.atherton
Post date August 2, 2025

Differential geometry makes heavy use of calculus in order to study geometric objects such as smooth manifolds. Just as differential geometry builds upon calculus and other fundamentals, the notion of differential forms extend the approaches of differential geometry and calculus, even allowing us to study the larger class of surfaces that are not necessarily smooth. We will motivate from calculus a brief brush with differential forms and their geometric potential.

Review of Vector Fields

Recall a vector field is a special case of a multivariable vector-valued function (taking in a scalar or a vector and outputting a set of multidimensional vectors). Often they are defined in $\mathbb{R}^2$ or $\mathbb{R}^3$, but can also be defined more generally on smooth manifolds.

In three dimensional space, a vector field is a function $V$ that assigns to each point $(x, y, z)$ a three dimensional vector given by $V(x, y, z)$. Consider our identity function

$$ V(x, y, z) = \begin{bmatrix}x\\y\\z \end{bmatrix}, $$

where for each point $(x, y, z)$ we associate a vector with those same coordinates. Generalizing, for any vector function $V(x, y, z)$, we visualize its vector field by placing the tail of the corresponding vector $V(x_0, y_0, z_0)$ at its point $(x_0, y_0, z_0)$ in the domain of $V(x, y, z)$.

Here we consider fluid particles (displayed as blue dots) as points, each with its own magnitude, direction, and velocity that correspond to overall fluid flow. Then to each point in our sample of particles we can assign a vector with speed of velocity as its length. Each arrow not only represents the velocity of the individual particle it is attached to, but also takes into account how the neighborhood of particles around it move.

Vector fields can be transformed into scalar fields via $\nabla$ taken as a vector differentiable operator known as the gradient operator.

Our gradient points in the direction of steepest ascent of a scalar function $f(x_1, x_2, x_3, . . . x_n)$, indicating the direction of greatest change of $f$. A vector field $V$ over an open set $S$ is a gradient field if there exists a differentiable real-valued function (the scalar field) $f$ on $S$. Then

$$V = \nabla f = ( \frac{\partial f}{\partial x_1}, \frac{\partial f}{\partial x_2}, \frac{\partial f}{\partial x_3}, . . . \frac{\partial f}{\partial x_n})$$

is a special type of vector field $V$ defined as the gradient of a scalar function, where each $ v \in V$ is the gradient of $f$ at that point.

The Divergence and Curl Operators

Now that we have reviewed vector fields, recall that we can also perform operations like divergence and curl over them. When fluid flows over time, some regions of fluid will become less dense with particles as they flow away (negative divergence). Meanwhile, particles tending towards each other will cause the fluid in that region to be more dense (positive divergence).

Luckily, we have our handy divergence operator to take in the vector-valued function of our vector field and output a scalar-valued function measuring the change in density of the fluid around each point. Given a vector field $V$, we can think of divergence

$$\text{div}\space V = \nabla \cdot V = \frac{\partial V_1}{\partial x} + \frac{\partial V_2}{\partial y}+ \frac{\partial V_3}{\partial z} . . . $$

as a variation of the derivative that generalizes to vector fields $V$ of any dimension. Another “trick” to thinking about the divergence formula is as the trace of the Jacobian matrix of the vector field.

In differential geometry, one can also define divergence on a Riemannian manifold $(M, g)$, a smooth manifold $M$ endowed with the Riemannian metric $g$ as a choice of inner product for each tangent space of $M$.

Every smooth manifold admits a Riemannian metric.

Whereas divergence describes changes in density, curl measures the “rotation”, or angular momentum, of the vector field $V$. Likewise, the curl operator takes in a vector-valued function and outputs a function that describes the flow of rotation given by the vector field at each point.

Here, all the points go in circles in the counterclockwise direction, which we define by $V(x, y) = \begin{bmatrix} -y \\ x \end{bmatrix} = -y\hat{i} + x\hat{j}$.

For a vector field function

$$V(x, y) = v_1(x, y)\hat{i} + v_2(x, y)\hat{j},$$

we can look at the change of a point $(x_0, y_0)$ by looking at the change in $v_1$ as $y$ changes and $v_2$ as $x$ changes. We calculate

$$2\text{d curl} \space V = \nabla \times V = \frac{\partial v_2}{\partial x}(x_0, y_0) \space – \frac{\partial v_1}{y}(x_0, y_0),$$

with a positive evaluation indicating counterclockwise rotation around $(x_0, y_0)$ and a negative evaluation indicating clockwise.

Approach by Differential Forms

Vector calculus can also be viewed more generally and geometrically via the theory of differential forms.

While curl is typically only defined in two and three dimensions, it can be generalized in the context of differential forms. The calculus of differential $k$-forms offers a unified approach to define integrands over curves, surfaces, and higher-dimensional manifolds. Differential 1-forms of expression form

$$\omega = f(x)dx$$

are defined on an open interval $ U \subseteq \mathbb{R^n}$ with $f: U \to \mathbb{R}$ a smooth function. We can have 1-forms

$$\omega = f(x, y)dx + g(x, y)dy$$

defined on an open subset $ U \subseteq \mathbb{R^2}$ ,

$$\omega = f(x, y, z)dx + g(x, y, z)dy + h(x, y, z)dz $$

defined on an open subset $ U \subseteq \mathbb{R^3}$, and so on for any positive integer $n$.

Correspondence Theorem: Given a 1-form $\omega = f_1dx_1 + f_2dx_2 + . . . f_ndx_n$, we can define an associated vector field with component functions $(f_1, f_2, . . . f_n)$. Conversely, given a smooth vector field $V = (f_1, f_2, . . . f_n)$, we can define an associated 1-form $\omega = f_1dx_1 + f_2dx_2 + . . . f_ndx_n$.

Via the exterior derivative and an extension of the correspondence between 1-forms and vector fields to arbitrary $k$-forms, we can generalize the well-known operators of vector calculus.

The exterior derivative $d\omega$ of a $k$-form is simply a $k + 1$-form. We provide an example of its use in defining curl:

Let $\omega = f(x, y, z)dx + g(x, y, z)dy + h(x, y, z)dz$ be a 1-form on $ U \subseteq \mathbb{R}^3$ with its associated vector field $V = (f, g, h)$. Its exterior derivative $d\omega$ is the 2-form

$$d\omega = (\frac{\partial h}{\partial y} – \frac{\partial g}{\partial z}) dy \wedge dz + (\frac{\partial f}{\partial z} – \frac{\partial h}{\partial x}) dz \wedge dx + (\frac{\partial g}{\partial x} – \frac{\partial f}{\partial y}) dx \wedge dy,$$

where $\wedge$ is an associative operation known as the exterior product. We define the curl of $V$

$$ \nabla \times V = (\frac{\partial h}{\partial y} – \frac{\partial g}{\partial z}), (\frac{\partial f}{\partial z} – \frac{\partial h}{\partial x}), (\frac{\partial g}{\partial x} – \frac{\partial f}{\partial y})$$

as the vector field associated to the 2-form $d\omega$. Attempt to see how this definition of curl agrees with the traditional version, but can be applied in higher dimensions.

The natural pairing between vector fields and 1-forms allows for the modern approach of $k$-forms for multivariable calculus and has also heavily influenced the area of geometric measure theory (GMT), where we define currents as functionals on the space of $k$-forms. Via currents, we have the impressive result that a surface can be uniquely defined by the result of integrating differential forms over it. Further, the differential forms fundamental to GMT open up our toolbox to studying geometry that is not necessarily smooth!

Acknowledgements. We thank Professor Chien, Erick Jiminez Berumen, and Matheus Araujo for exposing us to new math, questions, and insights, as well as for their mentorship and dedication throughout the “Developing a Medial Axis for Measures” project. We also acknowledge Professor Solomon for suggesting edits incorporated into this post, as well as for his organization of the wonderful summer program in the first place!

References

Post author By stephanie.p.atherton
Post date August 2, 2025

Research

Hidden Quivers: Supporting the Manifold Hypothesis

Post author By stephanie.p.atherton
Post date July 21, 2025

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

Post author By stephanie.p.atherton
Post date July 21, 2025
Tags Deep Learning, Geometry