Hodge Spectral Surrogates for Topology-Constrained Optimization

We first introduce the Betti numbers of clique complexes as topological quantities for graph data. Let $G=(V,E)$ be a graph, where $V$ is the set of vertices and $E$ is the set of edges. The clique complex $X(G)$ of $G$ is a simplicial complex obtained by regarding each clique in the graph as a simplex. That is, a subset of vertices

is a $k$-simplex if, for any distinct $i,j$,

holds. Equivalently, $\sigma$ must form a $(k+1)$-clique. Therefore, the clique complex is defined as

Let $C_{k}(X(G))$ be the $k$-th chain group of $X(G)$, and let

be the boundary operator. The $k$-th homology group is defined by

The dimension of this homology group, $\beta_{k}(X(G))=\dim H_{k}(X(G))$ is called the $k$-th Betti number. In particular, $\beta_{0}$ represents the number of connected components, while $\beta_{1}$ represents the number of independent one-dimensional cycles, namely loops that are not filled by triangles or higher-dimensional simplices.

The Betti numbers of clique complexes capture higher-order connectivity and cyclic structures of graphs. Therefore, controlling the Betti numbers of clique complexes in graph generation or network design is meaningful for controlling redundancy and robustness of graph structures.

Suppose that we want to optimize a graph so that the Betti number of its clique complex approaches a target value. For example, for a target Betti number $\beta_{k}^{\mathrm{tar}}$, one may consider the loss function

This formulation is natural from an intuitive viewpoint. However, it is difficult to optimize this loss function using standard gradient-based methods.

The reason is that the map

is combinatorial and discrete. A small change in the presence or absence of edges can discontinuously change the set of simplices contained in the clique complex. As a result, the ranks of boundary operators and the dimensions of homology groups can also change discontinuously.

Next, we introduce persistent homology[edelsbrunner2002topological, zomorodian2004computing, cohen2005stability, edelsbrunner2008persistent, otter2017roadmap] as a topological quantity for point-cloud data and related data types. Suppose that, from data $X$, we construct a sequence of simplicial complexes

according to a scale parameter. Such a nested sequence of simplicial complexes is called a filtration. For example, for point-cloud data, one can construct a filtration using Vietoris–Rips complexes indexed by a distance scale $r$.

At each scale $r$, the $k$-th homology group is

Persistent homology tracks when homology classes are born and when they die along the filtration. For a homology class $\gamma$, let its birth time be $b_{\gamma}$ and its death time be $d_{\gamma}$. Then the $k$-th persistence diagram is represented as

The persistence of $\gamma$ is defined by $\operatorname{pers}(\gamma)=d_{\gamma}-b_{\gamma}$.

Persistence diagrams and barcode diagrams provide compact summaries of the multiscale topological structure of data. Homology classes with long persistence are often interpreted as reflecting essential structures of the data rather than noise. For this reason, persistent homology is widely used for shape analysis of point clouds, images, time-series data, and related objects.

One way to incorporate persistent homology into machine learning is to construct loss functions from barcode diagrams or persistence diagrams. In general, given a target persistence diagram $D^{\mathrm{tar}}$, one may define a loss function for input data $X$ as

For example, if the goal is to remove undesired homology classes, let $U$ be the set of undesired classes. Then one can use a loss of the form

This loss acts to shorten the lifetimes of undesired homology classes.

On the other hand, if the goal is to preserve specific homology classes, let $T$ be the set of classes to be preserved, and let $\tau_{\gamma}$ be the target persistence. Then one can consider the loss function

Such barcode-based loss functions can be differentiated through birth and death values. The birth and death of each homology class $\gamma$ are determined by certain critical simplices. That is, one can write

where $\sigma_{b}$ is the simplex that determines the birth, $\sigma_{d}$ is the simplex that determines the death, and $f$ is the function assigning filtration values to simplices.

Then the gradient of the loss function can formally be written as

Thus, in barcode-based loss functions, gradients can be computed through the critical simplices that determine birth and death values.

This property makes it possible to use persistent homology as a loss function. However, this formulation has the following three important issues.

The preceding subsections identify two related but distinct uses of topology in data analysis and optimization. The first is descriptive: one computes Betti numbers, persistent homology, or barcode diagrams of a fixed data set in order to describe its global structure. The second is prescriptive: one seeks to optimize a point cloud, graph, or model output so that its topology satisfies a desired condition. This paper focuses on the second use.

The difficulty is that the topological quantities considered above do not directly provide smooth optimization signals. For clique complexes of graphs, the map

is combinatorial, because the set of simplices and the ranks of the boundary operators can change discontinuously when edges are added or removed. For Vietoris–Rips filtrations, barcode-based losses make persistent homology differentiable through birth and death values, but the resulting gradients may be localized on critical simplices, may change abruptly when persistence pairings switch, and may discard geometric and spectral information that could be useful for optimization.

Our goal is therefore not merely to compute topological summaries, but to construct differentiable objectives that retain a clear relationship with homology while providing smoother and more spatially distributed optimization signals. We use the Hodge-theoretic identity between homology and zero modes of the combinatorial Laplacian as the starting point. For a fixed finite simplicial complex $K$, the kernel of the $q$-th Hodge Laplacian $L_{q}(K)$ is naturally isomorphic to $H_{q}(K)$, and hence $\dim\ker L_{q}(K)=\beta_{q}(K)$.

The main question is how to use this identity when the simplicial complex itself changes with continuous parameters. In the next section, we address this question by embedding changing complexes into a fixed ambient chain space, separating inactive simplex directions by a penalty term, and then replacing hard simplex indicators with soft simplex weights. This leads to a Hodge-Laplacian-type spectral relaxation that recovers the ordinary Hodge-theoretic interpretation in the hard limit and provides differentiable low-frequency spectral quantities in the soft regime.

Let $K_{\max}$ be a finite fixed ambient simplicial complex containing all candidate simplices that may appear during optimization. For example, given a candidate edge set $E_{\max}$, one may define the maximum candidate graph

and use its clique complex $K_{\max}=X(G_{\max})$ as the ambient complex.

For each dimension $q$, define the ambient $q$-chain space by

This is a finite-dimensional real vector space generated by all oriented $q$-simplices in $K_{\max}$. After fixing orientations and equipping it with the standard inner product, $C_{q}^{\mathrm{amb}}$ can be regarded as a finite-dimensional Hilbert space.

Let $B_{q}:C_{q}^{\mathrm{amb}}\to C_{q-1}^{\mathrm{amb}}$ be the ambient boundary operator of the fixed complex $K_{\max}$. For an oriented simplex $\sigma=[v_{0},\ldots,v_{q}]$,

and the matrix representation of this operator is $B_{q}$. Since the boundary of a boundary is zero, we have $B_{q}B_{q+1}=0$.

Suppose that a hard subcomplex $K(\theta)\subset K_{\max}$ is determined by a parameter $\theta$. For each $q$-simplex $\sigma\in K_{\max}^{(q)}$, define the activity indicator by

The orthogonal projection onto the active $q$-simplex directions is defined by

Then the $q$-chain space of $K(\theta)$ is embedded as

Since $K(\theta)$ is a simplicial complex, every face of an active simplex is also active. Hence,

or equivalently,

We define the hard restricted boundary operator on the ambient space by

On the active subspace $C_{q}(K(\theta))=\operatorname{Im}\Pi_{q}(\theta)$, this agrees with the ordinary boundary map of the subcomplex $K(\theta)$.

A naive Hodge operator on the fixed ambient space $C_{q}^{\mathrm{amb}}$ would be

However, since this operator acts on the entire ambient space, inactive simplex directions may remain as spurious zero modes that do not correspond to homology classes of the active subcomplex. To remove them, we add an inactive-direction penalty. For $\mu>0$, define

The last term assigns energy $\mu$ to inactive $q$-simplex directions and removes them from the low-eigenvalue region.

With respect to the orthogonal decomposition

we have

Therefore,

Hence,

Thus, the number of zero modes of the penalty-regularized hard ambient Hodge Laplacian agrees with the $q$-th Betti number of the active subcomplex.

We now relax the hard projection $\Pi_{q}(\theta)$ using a soft graph and construct a differentiable spectral relaxation of the hard ambient operator. Let $V=\{1,\ldots,n\}$ be the vertex set, and let $E_{\max}\subset\binom{V}{2}$ be the set of candidate edges. For each candidate edge $e\in E_{\max}$, define an edge logit $a_{e}(\theta)\in\mathbb{R}$, and define the corresponding soft edge activation by

We call $G_{\theta}=(V,E_{\max},p_{\theta})$ a soft graph.

The activation or soft weight of a simplex $\sigma$ is defined as the product of the activations of its constituent edges:

Vertices are assumed to always exist, so $w_{v}(\theta)=1$. In particular, for an edge $e$, $w_{e}=p_{e}$, and for a triangle $t=\{i,j,k\}$,

In the hard limit $p_{e}\to\mathbf{1}_{e\in E}$, we have

Thus, this construction is a continuous relaxation of the ordinary clique complex at the level of simplex indicators.

For each dimension $q$, define

Using these matrices, define the weighted boundary-type operator by

In components,

Thus, the boundary relation between a simplex and its face is smoothly weighted by their activations.

We define the $q$-th penalty-regularized Hodge-Laplacian-type spectral relaxation by

The first term is the lower Laplacian-type term, the second term is the upper Laplacian-type term, and the third term is the inactive-direction penalty.

If $w_{\sigma}(\theta)\approx 1$, the penalty has almost no effect on the corresponding direction. If $w_{\sigma}(\theta)\approx 0$, the direction receives approximately energy $\mu$. In the hard limit $w_{\sigma}(\theta)\to\chi_{\sigma}(\theta)$, we have

Therefore, $\widehat{L}_{q}(\theta)$ is a differentiable Hodge-spectral relaxation of the hard Hodge Laplacian that recovers the Betti-number interpretation in the hard limit. Away from the hard limit, its low-frequency spectrum should be interpreted as a spectral surrogate for homological structure rather than as an exact Betti-number representation.

In graph generation or graph-structured optimization, the optimization variables are the edge logits

Then $p_{e}(a)=\sigma(a_{e})$ and

Thus, the Hodge-Laplacian-type spectral relaxation is obtained through

For controlling one-dimensional cyclic structure, which is measured by the first Betti number in the hard clique-complex limit, we use $q=1$ and define

This operator acts on the fixed ambient edge space $C_{1}^{\mathrm{amb}}$. Hence, graph optimization or graph-structured topology control can be performed in a fixed-dimensional Hilbert space, even though the underlying hard clique complex changes combinatorially.

We next represent Vietoris–Rips filtrations of point-cloud data within the same soft-graph and Hodge-spectral framework. Let

be a point cloud. In the ordinary Vietoris–Rips complex at scale $r>0$, an edge $(i,j)$ is present when $\|x_{i}-x_{j}\|\leq r$.

To relax this condition smoothly, define the stabilized distance with a small numerical parameter $\delta>0$

and define the Vietoris–Rips edge logit by

Here, $\varepsilon>0$ controls the softness of the threshold. The corresponding soft edge activation is

This value is close to one when $d_{ij}(X)<r$, and close to zero when $d_{ij}(X)>r$. In the hard-threshold limit $\varepsilon\to 0$,

The activation of a simplex $\sigma$ is defined by

In the hard limit,

because the Vietoris–Rips condition is equivalent to requiring all edges in $\sigma$ to have length at most $r$, namely $\max_{e\subset\sigma}d_{e}(X)\leq r$. Thus, the soft clique construction recovers the ordinary Vietoris–Rips complex at each fixed scale in the hard-threshold limit.

Thus, the Vietoris–Rips filtration can be represented as the soft clique complex of a soft graph whose edge logits are

For each scale $r$, we obtain the sequence

More explicitly, define

and

Then

This is the Vietoris–Rips instance of the Hodge-Laplacian-type spectral relaxation defined above. For controlling one-dimensional homology, we use $q=1$.

Persistent homology describes the birth and death of homology classes across scales. Therefore, instead of using a single scale, we consider a sequence

For each scale $r_{m}$, define

This yields a Hodge-Laplacian-type spectral relaxation

at each scale. The family

can be regarded as a Hodge-spectral smooth relaxation of the Vietoris–Rips filtration. In the next section, we apply low-pass spectral filters to these operators and define differentiable topological objectives for controlling Betti-type and persistent-homological structures.

Let

be the Hodge-Laplacian-type spectral relaxation, and let

be its eigendecomposition. In the hard ordinary-complex case, homological modes correspond to zero eigenvalues. In addition, near-zero modes with small positive eigenvalues may reflect geometrically meaningful structures such as almost-closed cycles, weakly filled holes, or unstable higher-order structures. Thus, it is useful to treat not only exact zero modes but also the low-eigenvalue region smoothly.

We call a smooth function $f:\mathbb{R}_{\geq 0}\to\mathbb{R}$ a low-pass spectral filter if it emphasizes low eigenvalues and suppresses high eigenvalues. The corresponding matrix function is defined by

In this work, we use the following representative filters.

The heat filter is defined by

where $\tau>0$ is a temperature parameter. It preserves zero modes and exponentially suppresses high-eigenvalue modes.

The resolvent filter is defined by

where $\alpha>0$. It also emphasizes low-eigenvalue modes, but its decay is rational rather than exponential.

The polynomial moment filter is defined by

where $\alpha>0$ and $d\in\mathbb{N}$. A zero mode contributes one, while high-eigenvalue modes are suppressed under suitable choices of $\alpha$ and $d$. More precisely, this polynomial behaves as a low-pass filter only when the positive spectrum is mapped inside the unit disk; this condition is made explicit below. Since this filter is expressed as a polynomial of the Laplacian, it does not require eigendecomposition and is compatible with matrix-vector products, stochastic trace estimation, and structural connections to possible quantum or quantum–classical hybrid computation.

As a more stable polynomial approximation, one may also use a Chebyshev polynomial filter. After normalizing the Laplacian by an upper bound or estimate of its largest eigenvalue, for example

so that its spectrum is approximately contained in $[-1,1]$, define

where $T_{\ell}$ is the $\ell$-th Chebyshev polynomial, $c_{\ell}$ are coefficients approximating a desired low-pass function, and $D$ is the polynomial degree. This filter can be computed recursively without eigendecomposition and is useful for approximating low-frequency spectral quantities in large-scale problems.

One way to use a low-pass filter is to compare low-energy subspaces themselves. Let $\theta_{\mathrm{tar}}$ be a target parameter and define

For a low-pass filter $f$, we define the spectral-filter matching loss by

This loss compares not only the number of zero modes but also the orientation and distribution of the low-eigenvalue subspace. Therefore, compared with barcode-based losses, it can use richer geometric and spectral information.

For the heat and resolvent filters, this gives

and

These losses act to move the current structure toward the low-energy subspace of the target structure. Although the heat and resolvent filters approximate the spectral projector onto the kernel in suitable parameter limits, they are not idempotent projectors for finite parameter values. For this reason, we refer to these objectives as spectral-filter matching losses rather than projector losses.

Another way to use a low-pass filter is to construct a smooth low-frequency spectral-mass surrogate for Betti-type information from its trace. For a low-pass filter $f$, define

If $f(0)=1$ and $f(\lambda)\approx 0$ for high eigenvalues, then $S_{f}(\theta)$ approximates the number of zero and low-eigenvalue modes and can be regarded as a smooth surrogate for $\beta_{q}$ in the hard ordinary-complex setting, and as a low-frequency spectral mass in the soft setting.

Given a target spectral-mass value $\tau_{q}$, define

In particular, for the polynomial moment filter, set

and define

For an ordinary hard complex, if

then

Thus, the parameters $\alpha$ and $d$ should be chosen together with an appropriate spectral scaling of the Laplacian. This loss moves the low-frequency spectral mass toward the target value without explicitly counting Betti numbers.

When graph sizes or ambient chain-space dimensions differ, directly comparing trace values introduces scale dependence due to the number of simplices. Therefore, let

and define the normalized moment

In the hard ordinary-complex setting, this normalization corresponds to comparing a Betti-type quantity per $q$-simplex. In the soft ambient setting, it should be interpreted as normalized low-frequency spectral mass. Given a target normalized value $\bar{\tau}_{q}$, define

This normalization yields topological losses that are more comparable across graphs or ambient complexes of different sizes.

In graph generation or graph-structured topology control, the optimization variables are the edge logits

The edge activations $p_{e}(a)=\sigma(a_{e})$ define a soft clique complex and hence a Hodge-Laplacian-type spectral relaxation $\widehat{L}_{q}(a)$. To control one-dimensional cyclic structures in the hard clique-complex limit, we take $q=1$ and use

Define the Laplacian moment and its normalized version by

where $N_{1}=\dim C_{1}^{\mathrm{amb}}$. The quantity $\overline{S}_{1,d}(a)$ can be regarded as a normalized $\beta_{1}$-type surrogate in the hard-limit sense, and as a normalized low-frequency $H_{1}$-type spectral mass in the soft setting. Given a target normalized spectral-mass value $\bar{\tau}_{1}$, define the graph topological loss by

In practical graph-generation tasks, one may combine this topological loss with ordinary graph objectives such as edge density, degree distribution, clustering coefficient, or task-dependent losses. If such a base loss is denoted by $\mathcal{L}_{\mathrm{base}}(a)$, the total loss is

where $\lambda_{\mathrm{topo}}>0$ controls the strength of topological regularization. This formulation allows one to optimize ordinary graph-structural objectives while controlling cyclic structures corresponding to $\beta_{1}$ of the clique complex. The ambient normalization above is the abstract formulation used in this section. In the numerical graph experiments below, we use an ordinary-complex or weighted-current-complex implementation and normalize by an effective edge count; the relation between these choices is discussed in Section 6.

Let

be a point cloud, and let

be a sequence of scales. For each scale $r_{m}$, define the edge logits by

These logits define a soft graph, a soft clique complex, and a Hodge-Laplacian-type spectral relaxation

at each scale. Since persistent homology describes the birth and death of homology classes across scales, the corresponding Laplacian-based losses are defined as sums over scales.