Bayesian Computation Meets Topology
Authors: Julius von Rohrscheidt
Bastian Rieck
Sebastian M Schmon
Venue: Accepted by TMLR ()

Abstract: Computational topology recently started to emerge as a novel paradigm for characterising
the ‘shape’ of high-dimensional data, leading to powerful algorithms in (un)supervised
representation learning. While capable of capturing prominent features at multiple scales,
topological methods cannot readily be used for Bayesian inference. We develop a novel
approach that bridges this gap, making it possible to perform parameter estimation in a
Bayesian framework, using topology-based loss functions. Our method affords easy integration
into topological machine learning algorithms. We demonstrate its efficacy for parameter
estimation in different simulation settings.

Published in Transactions on Machine Learning Research (11/2024)
Bayesian Computation Meets Topology
Julius Rohrscheidt
Technical University of Munich, Helmholta Munich
Bastian Rieck'
University of Fribourg, Technical University of Munich, Helmholtz Munich
Scbastian M. Schmon"
sebastaan..ccmmonn@@mmaacccc
Altos Labs, Cambridge, UK
Reviewed on OpenReview: https: // openreview. net/ forum? id= o4...
Abstract
Computational topology recently started to emerge as a novel paradigm for characterising The 'shape" of high-dimensional data, leading to powerful algorithms in (un) (un)supervised representation learning. While capable of capturing prominent features at multiple scales, topological methods cannot readily be used for Bayesian inference. We develop a novel approach that bridges this gap, making it possible to perform parameter estimation in a Bayesian framework, using topology-based loss functions. Our method affords easy integration into topological machine learning algorithms. We demonstrate its efficacy for parameter estimation different simulation settings.
1 Introduction
Topological machine learning methods enable describing the 'shape' of data at multiple scales while remaining impervious to many different types of noise. This led to strong hybrid models that combine geometry and topology in different domains, including computer vision (Hu et al., 2019; Waibel et al., 2022), graph learning (Horn et al., 2022; Yan et al., 2022; Zhao al., 2020), time series analysis (Zeng al., 2021), and representation learning (Carrière et al., 2020; Moor et al., 2020). Despite the advantageous properties of such integrations, topological methods cannot be readily integrated into Bayesian inference frameworks. However, the need lo deal with topological information following Bayesian paradigms frequently occurs in real-world data analysis tasks, for example whenever the shape of data is informative for the underlying inference task, or in cases where either uncortainty quantification is crucial or available data is not sufficiently large for approaches based on neural networks. such scenarios, assuming a known prior distribution p(0) and real- world data y. one could employ Bayes' formula to obtaín the posterior distribution p(Oly) ply mlvl)) 9//r))p(0). The posterior distribution offers multiple ways gain point estimators of the parameter, and allows for a comprehensive description of parameter uncertainties. Regrettably, the application of this formulation to complex generative processes presents substantial challenges, because the likelihood function p(y [0) and consequently, the marginal likelihood p(ツ) = f p(y 0)p(O)d0, frequently become analytically intractable. Moreover, the complexity in such systems requires methods that capture structural properties that are invariant to isometries (such as rotations and translations). making computational topology a well-suited tool (Hensel et al., 2021). Topology-driven Bayesian computation needs to satisfy two key requirements, namely (i) must effectively utilise topological descriptors in the observed data y, and (ii) it should offer a theoretically sound notion of posterior beliefs. Both criteria critical for overcoming the limitations of topological machine learning methods in complex scenarios and enhancing their applicability to rorldworld data analysis tasks, when the available amount of data is too small to utilise neural networks (c.g. due to
"These two authors contributed equally as last authors. i work carried while employed at Durham University.

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Published in Transactions on Machine Learning Research (11/2024)
(a)
(b)
Figure 1: Given a point cloud (a), persistent homology lets obtain shape descriptors known persistence diagrams (b). Due to their complex nature, the space of persistence diagrams is not directly amenable to Bayesian treatment. We overcome this limitation by using loss functions that characterise differences between observed and simulated samples.
expensive simulations), or when uncertainty quantification is important. To effectively articulate our posterior beliefs, we describe the scenario of this paper: In the absence of likelihood function, access to information the parameter 0 is solely available through samples from generative simulation models, which we denote by x p(1 0). Moreover, we do not assume p(x 0) to be differentiable in 0, preventing parameter inference via neural networks. This assumption is crucial for plethora of chaotic models such as collective motion, since such systems often exhibit tipping points that lead to highly non-continuous behaviour. Instead, we assume that we have access to a loss function of the form (z, y) + É(x, y), which measures to what extent a generated sample 'close' to the observed data y. In the general setting as outlined above, we demonstrate that the distribution T y) = ( @)p(0)da/ [f exp(=((g.=))((= ()p(d)dzde offers an alternative for encapsulating the inherent uncertainty in such generative models.
The fundamental challenge with this approach is to find loss function é that is suitable for the inference problem at hand. While traditional approximate Bayesian computation (ABC) often makes use of losses based on summary statistics derived from the given data, such statistics are not straigh to define, and usually far from being sufficient. By contrast, geometrical losses like the Hausdorff distance often do not capture the necessary information in order to perform accurate inference.
As the central contributions of this paper, we (i) leverage Bayesian inference and topology via efficient teometrical-topooogiaal loss terms that are capable of handling different data modalities, (ii) show that the comparison-based posterior, which we define in Section 3, can be seen as the solution to a generalised Bayesian optimization problem, (ii) demonstrate how sampling algorithms can leverage appologically-infommee loss functions to perform Bayesian inference, and (iv) we demonstrate the utility of our topology-aware approach on several complex generative models. Since our approach is Bayesian by nature, prior knowledge about the parameters to be inferred can readily be integrated into the workflow, making our workflow interpretable in comparison to black-box models.
In nutshell: Our framework fits into the realm of simulation based inference, where an exact likelihood function may not. be determined, but with a simulating model being accessible in order to sample data from the likelihood. The main novelty consists of incorporating topological information into parameter inference and showing its utility for parameter estimation in complex systems.
2 Background: Computational Topology
Before we introduce our approach, we first provide brief overview of relevant methods from computational topology. Such methods recently emerged, making use of ceometriealtoopolggiaall describe their overall 'shape! The concept of persistent homology is of particular interest in the context of machine learning since it leads to efficient descriptors of structured and unstructured data while at the same time satisfying invariance and robustness guarantees (Hensel et al., 2021).
topological properties of data to
Persistent homology obtains multi-scale goometrical combinatorial data structures. Given metric such as the Euclidean distance, persistent homology assigns family of shape descriptors, the persistence diagrams, to point clouds (Edelsbrunner & Harer, 2010);
shape descriptors of data by mcans of
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the process can be extended to other modalities. The calculation of persistence diagrams turns out to be stable with respect to the Hausdorff distance between point clouds, making this method highly robust in practice (Cohen-Steiner et al., 2007). See Fig. for an overview of this process.
The main idea is to obtain a nested sequence of simplicial complexes from the given data that encodes topological information about the data on malliple scales. The notion of 'scale' and how to define such a sequence in practice depends on the given problem; we will describe two constructions that are relevant in the context of this paper. Given point cloud X, the Vietoris-Rips compler at stage associated to X is the abstract simplicial complex in which a k-simplex is defined for every subset of X, consisting of k ++1 1 points in X that have diameter at most 2c. By varying c one obtains a sequence of nested simplicial complexes associated to X. The second construction is given by cubical complexes. Since we only use cubical complexes for greyscale images, we restrict ourselves to this particular setting, and refer to the literature for a more comprchensive introduction (Rieck et al., 2020b; Wagner et al., 2012). Any greyscale image A can be interpreted as graph, where the nodes are given by the non-zero pixels, such that there is an edge between two nodes if and only if the two nodes correspond to neighbouring pixels. For threshold e, we obtain new image from A by setting all pixels in A to zero, which admit pixel value less than e. Varying leads to a nested sequence of the graphs corresponding to the respective images, and therefore we obtain a nested sequence corresponding to A. Subsequently, we explain how to derive multi-scale topological features from such a nested sequence, also known as a filtration: Let 0 = Ko S K1 C ... C Km C Km = K be a nested sequence of simplicial or cubical complexes. The main idea of persistent homology involves tracking topological features, measured by means of homology groups, alongside this filtration. The family of boundary operators a(),, together with the inclusion homomorphism, induces a homomorphism between corresponding homology groups of the filtration, i.c. c² Ha(K,) + Ha(K)). groups for every dimension d. Given indices i S J. the dth persistent homology group is then defined as H3' == ker (K,)/ (im da+1 (Kg) n ker (k))). It can be seen as the homology group that contains all homology classes created in K, that are still present in Kj. Typically, the filtration of K has associated values ag S 01 S ..... 5m-115 S am (such as edge weights of a graph or pairwise distances of a point cloud). This permits us to summarise multi-scale topological activity in a persistence dingram: for each dimension d and each pair i < j, we store the pair (g) E R² with multiplicity Mij := (3.551-133 - 33,²)
This mapping yiclds sequence of homology
- -8 - 10--1-3311) -3- in a set (in practice, Mi.j = 0 for many pairs). The resulting set of points is called the dth persistence diogram Da. Given a point (x, y) E Da, the quantity pers(x, y) == ly - æ| is referred to as its persistence High persistence is commonly considered to correspond to features, while low persistence is seen to indicate noise Edelsbrunner & Harer (2010), but recent work shows that in many data sets, low persistence values rather indicate 'low reliability,' which may still play an important role for ream analyses (Bendich et al., 2016; Rieck et al., 2020b).
(d)
(d)
3 Methods
3.1
Bayesian Inference
Within the Bayesian framework, unknown quantitics, such as parameters denoted by 8, are furnishod with a probability distribution p(0), referred to the prior distribution. Subsequent. inference about the parameters 0 given observed data y is performed via the Bayesian update procedure. This process involves adjusting the prior distribution p(0) through the multiplication by the ratio of the likelihood to the evidence, i.c. p(0ly) = /()))1) However, applying this procedure in our particular context presents notable challenges. The primary issue stems from the fact that the likelihood function p(y ()) is typically unknown or analytically intractable, thus posing considerable difficulties for estimation, particularly when the data exhibits intricate goometrical geometrical-oppolgiiaal properties. To circumvent this problem, recent research has shown that the use of the likelihood function of the data can be avoided, producing so-called generalised Bayesian posteriors (Bissiri et al., 2016). One approach to construct such a generalisation to the Bayesian posterior can be found through the lens of viewing Bayesian inference as the solution to an optimisation problem (Csiszár, 1975; Donsker
For more in-depth discussion of computational topology, in particular in the context of machine learning, refer the reader Hensel al. (2021). 2A hegh-level explanation of homology groups is contained in the appendix.
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& Varadhan, 1975; Zellner, 1988). Specifically, it can be seen finding distribution q that balances prior information as measured by the Kullback-Leibh (KT.) divergence between the prior and the posterior candidate (which we want to minimise) and the expected log- log-likelihood of new observations y (which we want to macimise):
arg migmingerrn PEP(O)
+ KT.((()):
The solution to this problem is the traditional Bayesian posterior q-(d) (=(0)=p(0ly) Recently, generalisations of this approach have been suggested, demonstrating that. coherent beliel updates can also occur instead of the negative log-likelhhodd loss, - log p(y 0) any loss (y, 0) is taken (Bissiri al., 2016; Knoblauch al., 2022). The solution to the concomitant optimisation problem
arg (pP() {Egli 0)] + (L()
(1)
is given by
(0) exp(- 0) (0)
(pp--(y,0))(( exxp(=((y.0))
Tomi (0 (1))
(2)
However, the problem is that even in this general formulation, the loss formally scores observed data y with respect to parameter 0. A common approach is to use expectations over samples from a generative model, i.e. simulate x ~~ f(x 0) and approximate
1 Σ N
0) Ep(rl [e(y, x)]
Ely
(3)
with x; ~ p(x 0). Replacing the loss I(y, = Fppt (0) [Kyy x)]
with a sample average, however, changes the posterior from Eq. (2). We will demonstrate in the following that we can directly construct valid posterior from the loss function f(y,z).
3.2 Comparison- Based Posteriors
Uncertainly estimates predicated on distance functions have long-standing history in Bayesian statistics as means of approzimating posterior distributions, with notable use in the realm of approzimate Bayesion computation (ABC) (Tavaré et al., 1997; Pritchard et al., 1999; Beaumont et al., 2002). In the ensuing discussion, we will illustrate that modification of the aforementiones approach, specifically, generalised posterior from ABC such as Schmon et al. (2021), can furnish us with theoretically sound form of posterior update. The proposed approach involves using a loss function (x, y) + Ely, x) to construct a joint posterior involving simulated data x ~ p(# () and real data y. It is of the form
x (0, x lv) x exp (-ely, p(r ())()).
(4)
This approach, however, encounters several obstacles. Firstly, it is not clear that Eq. (4) defines reasonable notion of uncertainty. In addition, since æ is just simulated data, not observed data, we need to average over possible values, leading to
<001)) = fx exp p(r ())
(5)
f.xxx exp ((),))))) p( 0 (0))ddd
It turns out that this is is a valid (gencralised) posterior under the loss associated with the topological properties of our data, namely:
Proposition 3.1. The comparison-based posterior
exp (-p(((,)xx) p(a 0)p(()
x =0,, x
(6)
Jx Se exp (-E(y, x)) p(x ( ())(Or
3hhis occurs even in the case where the estimator unbiased since this property lost under the non-linear (exponential) transformation by virtue Jensen's inequality.
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is the solution (0,))) x(0, 1y) to the optimisation problem
arg (E,1((y, + KL (((,))0)))))
(7)
where the arg min ranges over all q E P(O X).
The proof can be found in the appendix. In Eq. (7), the first term in the objective function measures the expected loss under the posterior, where the loss function ((y,z) quantifies the discrepancy between the observed data and the simulated data z. This is precisely where we can use improved inductive biases or domain knowledge and employ loss functions such as the topological losses introduced in Section 3.4. The second term encourages the posterior to stay close to the prior, as in the traditional Bayesian setting, however, with a significant difference: the prior distribution p(0) over the parameters 0 together with the simulation model p(æ 0) naturally implies prior belicfs over what the data should look like. Thus, it sensible consider the KI-ddvergence between the joint beliefs p(ar ()p(O) and (x,,0).
Proposition 3.1 shows that the comparison-basee posterior of Eq. (4) is in fact solution to the respective optimisation problem (which is the analogue of Eq. (1) in our setting). The motivation is therefore to use comparison-based posterions for approximate inference under the assumption that an appropriate loss function is available. Proposition 3.1 does not necessitate a topological loss function and can be considered independently of specific loss function. However, in particular, permits incorporating topological information via topological losses.
To the best of our knowledge, the argument and proof that comparison-based posteriors can be as the solution to generalised Bayesian optimization problem is new. While distance-based loss functions, where parameter 9 is only available through simulation have been used before, they are commonly considered approximations of an idcalised loss function
e(y, 0) = lim
x:).
(8)
Thus, Proposition 3.1 demonstrates that comparison-based posteriors do not have to be seen as 'subooptiaal approximation to an abstract ideal, but can be seen valid generalised posterior in their right.
As such, our approach can be seen as generalisation of the traditional Bayesian posterior in the sense that it allows for a more flexible modelling of the data generation process. Instead of assuming that the observed data is generated exactly according to the model p(yl 0), we allow for the possibility that the data is generated from a distribution that produces data close to y in terms of the loss function t. This makes our approach potentially more robust to model misspecification and more suitable for complex data with intricate topological properties.
3.3 Inference for Comparison- Baseed Posteriors
Evaluating posteriors such as those given by Eq. (5) analytically is typically infeasible. An alternative approach may involve direct targeting of a variational distribution as shown in Eq. (7). However, this strategy encounters two main difficulties: firstly, the distribution p(r) ()) is often intractable, rendering the computation of the KL divergence term non-trivial. Secondly, in order compute the gradients, we must propagate them through the simulation models. This poses challenge as p(z (6) often represents 'black-box' model, not necessarily implemented within framework that supports automatic differentiation. In fact, many complex simulation models are not even differentiable in the parameters to be optimised. Additionally, such algorithms lypically require large amounts of data, which is a frequent limitation when simulations are computa anpputaaiinnal ionally expensive. Consequently, this typically precludes the practical application of gradient- based optimisation techniques. However, turns out that Monte Carlo algorithms, such as self-normalised importance sampling or so-called pseudo-marginal Markov chain Monte Carlo (Andricu & Roberts, 2009) are able to compute expectations with respect to posteriors of the form in Eq. (5).
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Algorithm Importance sampling estination of the mean
1: Input: Observed data test function h
2: for i == 1: do
3: Sample 0g ~ p(0)
4: Sample 2; ~ p( 0,)
end for Output:
E 2im1 h(0,)e-wellg
M(0) Ei=1 E wi(gga))
Algorithm 1 shows self-normalised importance sampling procedure. Almost-sure convergence can be shown under mild regularity conditions (e.g. Owen, 2013, Theorem 9.2) for test function of interest h, i...
Σ 1(0:)e-(uv)))
(9)
EE ((vie)))
approaches
IJ h(0)e f(v.z) p(x ()p(0)drde
If e c-(0)=))px) (0)(())ddd drde = E is [((())]
(10)
as n - oo. An alternative to use Markoe chain Monte Carlo (MCMC), where we produce a Markov chain converging to the distribution (5); see Algorithm 2 in the appendix. Such algorithms indeed converge to the desired target distribution (Andrieu & Roberts, 2009, Theorem 1) under mild assumptions. A high-level overview of pseudo-marginal MCMC is contained in the appendix.
3.4 Losses Based on Geometry & Topology
Having introduced multiple inference algorithms and explained how to obtain topological features, we now discuss how to derive vopology-based loss functions. We start with discussing metrics in computational topology. It turns out that persistence diagrams can be endowed with a metric by using optimal transport. Given diagrams D and D' containing features of the same dimensionality, their pth Wasserstein distance is defined
(,(DDD) :=
inf D-+D'
(11)
ZED
where n(-) denotes a bijection. Since D and DD generally have different cardinalities, we consider them contain an infinite number of points of the form (T, i.e. tuples of zero persistence; this akin to requiring each diagram to contain the projections of points to the diagonal, originating from the other diagram. A suitable v(-) can thus always be found. Solving Fq. (11) is practically feasible using modern optimal transport algorithms (Flamary et al., 2021). While this is the most 'principled' approach in the sense that such a loss formulation forms a proper metric in the mathematical sense alternative formulations to Eq. (11) exist, and our framework is fundamentally compatible with all of them. Along these lines, other topological descriptors, such Betti curves (Rieck al., 2020a) or persistence images (Adams et al., 2017), might be used instead of persistence diagrams. These descriptors are often easier to compute and afford the use of fast Lı or L2 distances proxies for Eq. (11). For instance, the L2 distance between persistence images is known to share some advantageous properties with the Wasserstein distance (Adams et al., 2017) but it only handles 'discretised' version of the data, so some information is invariably lost (or, to briefly consider an optimistic point of view, the persistence image is smoother than a persistence diagram). As another alternative to the previiusly-describee descriptors, we could also employ a kernel, ie. a similarity measure between persistence diagrams. These similarity measures are not metries the mathematical sense, lacking the requirement of the identity of indiscernibles. but are easier compute (unlike the persistence images, they do not require vectorisations) and require fewer parameters (Kwitt et al., 2015; Reininghaus et al., 2015).

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A loss function based on the Wasserstein distance. To leverage the power of computational topology and Bayesian inference, we propose combining the Bayesian framework for updating posterior probability estimates with a topological loss function. Given observed data y and simulated data a sampled from p(r 0), let D2 and Dy be the persistence diagrams corresponding to I and y, respectively. We then define a topological loss funclion er: Y xxx= - R, by setting
(v,x)) :.pt):=><((pp.) Wp(Py. D2).
(12)
This formulation applies to different data modalities. If the data can be represented as a point cloud, we calculate persistence diagrams from the Vietoris- Rips complex. For (greyscale) image data, we obtain persistence diagrams via cubical compleres, enabling topological feature calculations for more complicated modalitics such as MRI data Rieck et al. (2020b). As we outlined above, other choices for the loss function would be possible, and we may now continue with our Bayesian updating procedure. Notice that unless otherwise mentioned, we use p = to compute the topological loss.
Morcover, note that the implementation of giotto-tda (Tauzin et al., 2020), which we use in our experi- ments, uses faster approximate algorithm rather than the actual loss. Under the hood, this implementation uses an auction algorithm with a worst-case computational complexity of O m log((-))), with n and m being the number of points in the respective persistence diagrams and being a precision parameter.
The Wasserstein distance between persistence diagrams is stable in the sense that the geometric distance is an upper bound of the topological distance. Given two point clouds Y, X of the same cardinality, we have
Properties.
W, WD,,,, D.) S C, (Y,),,
(13)
where Cp refers to a constant that only depends on and W,,Y X) is defined similarly to Eq. (11), but calculated on the point clouds themselves Skraba & Turner (2022). Since (p(::) S (::):) for p' S p. low values of p are desirable in terms of stronger stability; we find that p = 2 provides a suitable compromise solution. Moreover, our loss function &r is invariant under isometries (Edelsbrunner & Harer, 2010) and stable under subsampling (Moor et al., 2020). Eq. (13) can be interpreted as follows: once X and Y are geometrically close, their corresponding topological properties are similar as well. However, the converse is not Irue in general since structural similarity with respect to topological properties may occur between two datasets although they are not close geometrically. In fact, the left-hand side of Eq. (13) is often substantially smaller than its rinht-hand side, which serves as the initial motivation to use topological losses instead of their geometrical counterparts, particularly for inference tasks in which the shape of data is informative for the underlying objective. We validate this theoretical observation empirically in Section 5, by showing that topology encoded in losses indeed lead to more accurate parameter inference.
A loss function based on the Hausdorff distance. As a baseline and computationally simpler comparison partner, we also define a geometry-based loss function based on the Hausdorff distance between point clouds X and Y. Given metric space (M, d) and two non-empty subsets X, Y C M, define our loss as
lx Y) = inf(e 2 v.
(14)
where X. = Umexfm E M; d(m, x) < c} denotes the e-thickening of X in M. While this loss does not satisfy invariance properties, it more efficient to compute in practice. However, as we will see in the experimental section, its utility in complex data generation scenarios is limited. For |X] n and |Y| m, the computational complexity of the Hausdorff distance is
4 Related Work
A range of Bayesian inference methods have been developed to address the challenges involved with uncertainty quantification when dealing with complex models whose likelihoods are analytically intractable. While each of these approaches offers distinct solution, they also present their own unique sets of challenges and limitations.
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Approximate Bayesian computation (ABC) is well-established approach (Tavaré et al., 1997; Pritchard et al., 1999; Beaumont, et al., 2002). ABC employs a pre-defined threshold to accept parameters that yield a small enough distance between the simulated and observed data. More recently, Wasserstein ABC (Bernton et al., 2019) has capitalised on optimal transport theory to define a distance function between simulated and observed data. This form of rejection-basel ABC comes with several potential problems such as the choice of the rejection threshold: a small threshold will lead to high numerical costs whereas a. threshold too large can result in incorrect coverage of credible regions (Frazier et al., 2018). To make matters worse, rejection ABC algorithms are known to exhibit non-standard asymptotic behaviour in case of model misspecification (Frazier et al., 2019).
Several authors have suggested constructing a generalised posterior along the lines of Eq. (2) using distance functions, including mazimum mean discrepancy (MMD) (Park et al., 2015; Chérief-Abdellatte & Alquier, 2020). However, these methods, in contrast to our approach; use a theoretical loss, for instance, I(v. = Ep(a [((y,))) which is then approzimated using empirical averages, thus potentially introducing a source of bias or instability. By contrast, our approach; is more related to (particle) filtering (e.g. Doucet et al., 2009), where inference is carried out over the latent state simultaneously, Miller & Dunson (2018) introduced a discrepancy-based Bayesian procodure they termed 'coarsening," where the observed data is considered as a coarsened version of latent data, thereby enhancing robustness.
Finally, topological data analysis (TDA) has already started to see some use in the context of analysing generative models. Some works are concerned with fitting the parameters of previously-defined prior distributions of persistence diagrams (Maroulas et al., 2022; Oballe, 2020), thus demonstrating the expressivity of TDA for describing complex systems. We are interested in parameter estimates of black-box models by exploiting this expressivity. Along these lines, Topaz et al. (2015) use persistent homology for a qualitative analysis of the topological characteristics of swarming models (such as Viesek model). In comparison, our approach is intended to leverage topology and Bayesian inference for quantitative studies, enabling parameter ostimation. More recently, Thorne et al. (2022), used topological statistics in combination with rejection sampling. This method constitutes the first use TDA in the set ting of ABC; it involves numerous parameter choices that our method does not necessitate. Moreover, the rejection sampling approach is itself parameter-driven and finding suitable rejection threshold requires additional domain knowledge. Our approach stands out by integrating loss function to construct posterior distribution directly, without resorting to approximation, thereby offering a theoretically-gronnde form of posterior belief. In comparison to the aforementioned method, ours does not necessitate the choice of topological summary statistics but operates on the full persistence information.
5 Experiments
We empirically validate the utility of our proposed method in the form of three main experiments that compare topological losses and respective geometrical losses, by using the results from Section 3. Tn the first two experiments, we focus on parameter inference for chaotic simulation models, which can be seen as (approximately) solving inverse problem. Notice that these are well-suited scenarios for applying topology-driven Bayesian methods, since chaotic models are known to admit likelihood functions that are neither analytically tractable (due to their complexity) nor differentiable (due to the presence of tipping points, for instance), thus precluding the use of both neural networks and traditional approaches to Bayesian inference. Moreover, our method requires only a very small number of simulations (here, 250) to perform reasonably, making our approach superior to neural networks with regards to the required number of data points (we find that this is particularly relevant in setting where the simulation is (computationally) expensive). Furthermore, use 100 observation points for inference, and normal distributions for both
for each run of the experiments, the prior and the proposal distribution in the MCMC procedure. The weight parameter w in the importance sampling procedure is set to 10. The third experiment involves a synthetic model that shows how our method may be used for inference in the context of imaging data. In all of our experiments, topological losses perform significantly better than a respective geometrical loss. Finally, we compare our results to standard summary statistics that are frequently used in the context of approximate Bayesian computation, see Appendix A. Remarkably, our topology-driven approach is observed to perform significantly better than Lo-norms on summary statistics for the first two non-synthetic experiments, underpinning our proposed method's utility
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Figure 2: Left to right: simulations of the Viesek model for 7) 0.15, 0.3 and 0.6, respectively. For lower values of 17 swarming behaviour emerges, whereas for higher values of swarms tend to merge and the model becomes more ergodic.
for practitioners. As a consistency check, we use our proposed methods for inferring the radius parameters of two objects, namely the 2-sphere and the 2-torus. Table in the appendix depicts the results of the inference process.
5.1 Vicsek Model:
Swarm Behaviour
The Viesek model is an agent-based model that is particularly useful to study collective motion like swarming and flocking. The underlying idea is that each agent aligns at any time step with its nearest neighbours, before the aligmment is perturbed by an additional random term. The resulting angle of the alignment determines the direction of movement for the subsequent time step, where the movement takes place with constant speed which is independent of time and individual. Following Viesek et al. (1995), the formula for the aligmment angle update of an individual i is given by 0( (0((),, + A0, where (0(t)), denotes the average of the angles of individuals with distance to agent at most and AO is uniformly sampled from the interval [-//2, 9/2] for 0 S 17 S 2T. The new position of the agent is then determined by moving with constant speed towards the direction of the updated angle. Notice that although the position of cach agent can be expressed analytically, the dependencies between agents leads highly complex behaviour the overall system over time, which cannot be expressed analytically. We therefore are in the proposed setting of an intractable likelihood, necessitating simulation-based approach to inference.
To demonstrate the utility of topological losses for parameter inference in setting where the respective likelihood function is not analytically tractable, we infer the noise parameter after a certain number of iterations (time steps) of the model. We distribute 2000 agents across a square of edge length L = 25, where opposite edges the square are identified with the orientation being preserved; the 'world' thus constitutes the surface of 2-torus. For the inference of n. we use 250 simulations for each sampling method, aiming to infer 77 after 5, 10, and 50 iterations (time steps) of the model, respectively, repeating the inference times. Since the Viesek model gives rise to point clouds, we calculate Vietoris- -iiriss--iisss complexes and compare the corresponding persistence diagrams. As baseline, Fig. 3 depicts the results. In our experiments, the results for the topological loss with the MCMC sampling procedure outperforms in all of the settings, and leading to highly accurate estimales even after 50 time sleps. By contrast, we note that the estimators that were calculated with the Hausdorff loss lead to particularly poor results the regimes that are closer to being ergodic, i.c. for larger values of 17. Finally, we remark that our approach with the proposed topological loss outperforms standard rejection sampling in combination with summary statisties; Appendix A.
the Hausdorff distance between the point clouds;

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(a) Importance sampling
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10 50
(b) MCMC sampling
Figure 3: Summary of parameler estimates for the noise parameter of the Viesek model, using 5 repelitions of the inference procedure. Each subplot depicts estimates for {0.15, 0.30, 0.60], respectively. The true parameter is shown as solid black line. The horizontal axis shows the number of iterations of the Viesek model. Estimates based on the geometrical loss la are less accurate than the topological loss ly, in particular for larger number of iterations. Please refer Table for the raw values.
(a) low Lurbulence
(b) medium turbulence
(c) high turbulence
Figure 4: Left to right: simulations of the fluid model after 3000 iterations, for 1 0.2, 0.3 and 0.4, respectively. For lower values of the fluid flow is less turbulent than for higher values of
5.2 Fluid Dynamics: Lattice Boltzmann Simulation
The lattice Boltzmann methods (LBM)
collection of algorithms that are used for fluid simulation. Instead of the conservation of macroscopic propertics of the dynamical system, LBM models the fluid over a discrete lattice, by performing local propagation and collision processes, for all particles simultaneously. For a thorough introduction, see Krüger et al. (2017). After initialising the state each particle equally, a portion of normally distributed noise added each state, independently. This portion of randomness is controlled by a global parameter 17 2 0, where higher values of n correspond to higher portion of noise in the initialisation. The crucial observation is that the degree of turbulence in the system be modelled by
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Table Estimating the randomness parameter the lattice Boltzmann model after 3000 simulation iterations, using repetitions of the inference procedure. The topological loss together with importance sampling outperforms all of the other configurations.
Randomness parameter "
Loss Sampling method
0.20
0.30
0.40
Importance sampling
0.16 + 0.02
0.22 + 0.04
0.35 + 0.13
la
MCMC
0.15 + 0.10
0.91 + 1.64
0.38 + 0.25
Importance sampling
0.21 + 0.01
0.27 + 0.00
0.38 - 0.01
er
MCMC
2.28 + 1.40
1.71 + 0.81
2.35 + 1.10
Table 2: Estimating the parameter the Percolation model, using 5 repetitions the inference procedure. The topological loss together with either importance sampling or MCMC sampling outperforms all other configurations.
Randomness parameter P
Loss
Sampling method
0.15
0.30
0.60
Importance sampling
0.18 + 0.00
0.30 1 0.47
0.59 1: 0.00
lg
MCMC
3.18 + 2.55
1.30 + 0.47
2.18 + 1.47
Importance sampling
0.15 + 0.00
0.29 + 0.00
0.59
+ 0.01
Cr
MCMC
0.15 + 0.02
0.29 - 0.00
0.59 + 0.00
varying 7, as is shown in Fig. 4. Our objective is therefore to infer the parameter 7) which corresponds to a given observation after a certain amount of itcrations in the model, where we fix this number of iterations be / 3000 time steps. As the topological loss, we use the Wasserstein distance between the persistence diagrams of the cubioal complezes corresponding to the observed and the simulated state, respectively, while the Hausdorff distance between the states serves baseline comparison. Table 1 shows the results; we observe that the topological loss together with importance sampling outperforms the geometrical loss in both configurations, by lar. Again, rejection sampling in combination with summary statistics performs less reliably; see Appendix A.
5.3 Percolation Model: Multi-Scale Structures
In statistical physics, percolation refers to the behaviour of network when links are added to it. Percolation can be viewed as a process of geometric phase transition since at critical state of the network, disconnected components merge into large connected clusters by the addition of a small fraction of links. In our setting, the percolation model is probabilistic in the sense that links are added randomly, with some probability p. and the goal is to infer p. We realise such a network as a square 2D greyscale image of pixel size n², for some positive integer n.4 We assign a value v to each pixel in the image, where is sampled uniformly from the set {1,..., greyscale value, which we set to 50 for our experiments. For the inference of p, we used 250 simulations for both the importance sampling and the MCMC sampling. Since we are dealing with greyscale images, we obtain persistence diagrams via cubical complexes.
with probability 1- p, and v with probability p. Here, Umax is the maximum realisable
We compare our proposed topological loss l to several other losses that are commonly used in imaging processing, including MSF, RMSF, the universal image quality index (UQT), the relative average spectral error (RASE), the spatial correlation coefficient (SCC), and the pixel-based visual information fidelity (VIFP). Since we find that the results of SCC and VIFP outperform other scores, and since SCC and VIFP lead to
Note that alternatively, greyscale image can be interpreted as an undirected graph, defining the of vertices be the of pixels, and adding edges between neighhouring pixels if both of their values non-zero.
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comparable results, we only show the results for SCC (see Zhou et al. (1998) for construction of SCC) in comparison to the ones obtained by the topological loss. Table 2 shows the results. We again observe that, our topology-based loss fy, together with MCMC sampling, outperforms the other methods by far. 5
However, in this particular experiment, rejection sampling together with a mean statistic performs even better. This is due to the idiosyncratic behaviour of the percolation model, which enables inferring the true parameter from the mean of pixel values; see Appendix A for an extended discussion. In all of our experiments, we observe that topological losses outperform a respective geometrical loss. This observation highlights the utility of topology in simulation- based inference: In particular complex systems exhibit typical collective behaviour, which can be captured by their topological descriptors (i.c. their 'shapc'), and which cannot be captured via geometrical losses, since geometric properties of the output of a simulating model (such as the ezact positions of individual agents) may differ even for a fixed parameter. In this sense, topological properties are often more stable with respect to the input parameters than geometrical properties, making topology potentially well-suited in a wide range of inference problems.
6 Discussion
We presented a novel Bayesian approach that allows for leveraging topological information in parameter estimation. Our method, which builds on the framework of generalised Bayesian inference, introduces a topology-based loss function into the construction of the posterior distribution. The resulting methodology is particularly suited for scenarios in which one has deal with a 'black-box' likelihood function, iee. where the likelihood is only accessible through simulations. This approach offers theoretically grounded form of posterior belief that improves upon some of the challenges inherent in existing distance- based methods, which rely on geometrical features. The main limitation of our approach is that, although computational topology appears to be the appropriate tool for certain complex models, the resulting performance for parameter estimation highly depends on the setting (Lueckmann et al., 2021). A careful assessment of the given scenario is recommended beforchand. We restrict ourselves to scenarios with low-dimensional parameter spaces this work, and leave the investigation of high-dimenssonal parameter spaces for future work. Moreover, we do not claim that our methodology outperforms benchmark Bayesian approaches under the assumption of a known likelihood function. However, we have empirically shown that our method outperforms geometrical approaches and summary statistics for parameter ostimation in the sctting of complex systems, which suggests future applications in other complex scenarios arising from life sciences data, for example.
Our work contributes to the ongoing evolution of Bayesian inference methods for complex models. It presents a new avenue of exploration that merges the advantages of both topological data analysis and Bayesian statistics. We believe that the principles and techniques introduced in this paper have broad applicability and offer compelling new approach for uncertainty quantification in wide range of data analysis tasks. From a machine learning perspective, the main novelty of this paper is to introduce framework that fuses topology and Bayesian inference for scenarios in which the application of gradient- gradient-based approaches like neural networks constitutes a major obstacle. We anticipate future research to focus on further refining this method, specifically when it comes to calculating efficient estimates in sparse regimes. Moreover, we plan on assessing the performance arising from other lopology-basee formulations ions, which are geared towards specific modalities like meshes (Turner et al., 2014) or time series (Zeng et al., 2021).
Reproducibility
We provide code for the given method and of the experiments shown in this paper under http://tttub..omm///ddsslll/////
Acknowledgements
B.R. was partially supported by the Bavarian state government with funds from the Hightech Agendo Bavaria.
Forr the parameter value of 0.3, we use the MSE loss performance measure, which is e 0.001 for each of the topological losses, and 0.211 for the geometrical loss.
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Table 3: Estimating the radius parameters 2D sphere (r) and a 2D torus (r, R). For the more complex torus parameter estimation, we observe that our topologiaal loss e provides more reliable estimates than the geometrical loss lg. The results show the mean of five runs for each parameter choice.
Sphere radius
Torus radii (r, R)
Loss Sampling method
5.00
(1,2)
(3, 5)
(5, 10)
Importance sampling
0.97
4.94
9.91
0.94
1.96
2.99
4.95
4.98
9.98
fg
MCMC
1.00
4.99
10.00
0.99
2.04
2.77
4.96
5.01
10.00
Importance sampling
1.01
5.02
9.98
1.02
1.96
3.01
5.03
4.95
10.04
Fr
MCMC
0.99
4.99
10.00
1.04
1.89
2.99
4.97
5.01
10.04
A
Additional results
In the interest of reproducibility and to improve the understanding of our method, also provide more detailed insights into experiments."
Synthetic data parameter estimates. As consistency check and illustrative example, we use our proposed methods for inferring the radius parameters of two objects, namely the 2-sphere and the 2-torus. The 2-sphere is the surface of a closed 3-dimensional ball radius r, while the 2-torus is constructed using an inner radius r and an outer radius R. The point clouds were sampled uniformly from the respective surface of known parameter.
Table 3 depicts the results of the inference process. We observe that both types of losses are capable of inferring the right parameters of these simple geometric objects, with our topology-based loss fr providing slightly more reliable estimates in the case of torus. Table 3 shows the raw values of learning radius parameters for synthetic data sets (2D spheres and 2D tori). This experiment primarily shows that for such simple shapes, geometry-based and topology-basee losses perform equivalently. For more complex data sets, however, we find that the improved robustness of our topology- based loss term helps inferring the ground truth parameters.
Swarm behaviour of the Viesek Model. Fig. 5 shows simulations of the Viesek model for different. noise parameters The higher the more 'chaotic' the behaviour of the resulting complex system.
Figure Left. to right: simulations of the Viesek model for v 0.15, 0.3 and 0.6, respectively. For lower values of swarming behaviour emerges, whereas for higher values of swarms tend to merge and the model becomes more ergodic.
Certaii parts of this section may have been developed with the OpenAT (2024).
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0.25
0.7
0.4
0.2
0.0
0.3
0.5
0.1
0.2
5
10
50
5
10
50
5 10 50
(a) Rejection sampling
Figure 6: Summary of parameter estimates for the noise parameter 7 of the Viesek model with respect to summary statistics and a standard rejection sampling procedure, using 5 repetitions of the inference procedure. Each subplot depicts estimates for E {0.15, 0.30, 0.60), respectivcly. The true paramcter is shown a solid black line. The horizontal axis shows the number iterations of the Vicsek model. Estimates based on the mean statistic Le loss and standard deviation statistic L2 loss both are significantly less accurate than the respective topological loss, see Fig. 3b.
Vicsek model parameter estimates. Accompanying Fig. 3, we show the raw parameter estimates n, the noise parameter the model in Table 4. Moreover, we show the results with summary statistics (mean and standard deviation, respectively), in Fig. 6. In the latter setting we used the L2 distance between the respective summary statistics as the loss that determines the results of our method. We observe that there is no statistic that clearly discriminates the other one, and that the performance of the results highly depends the parameter and time step that is considered. We find that the overall accuracy and reliability of the respective experiments by using topological losses is significantly higher; see Fig. 3 for comparison.
rejection sampling procedure
Table Noise parameter estimation after n {5, 50) iterations of the Viesek model. Estimates n based on a topological loss are always closer to the ground truth value than estimates obtained via a geometrical loss.
True parameter
Loss
Sampling method
0.15
0.30
0.60
Importance sampling MCMC
0.17
0.35
0.67
Hausdorff distance
0.13
0.36
1.15
n 5
Importance sampling MCMC
0.14
0.29
0.59
Topological distance
0.12
0.33
0.63
Importance sampling MCMC
0.11
0.39
0.70
Hausdorff distance
0.18
0.41
1.95
n 10
Importance sampling MCMC
0.18
0.27
0.65
Topological distance
0.13
0.34
0.62
Importance sampling MCMC
0.12
0.29
0.72
Hausdorff distance
0.28
1.40
2.28
n 50
Importance sampling
0.16
0.31
0.61
Topological distance
MCMC estimate
0.21
0.24
0.59
Lattice Boltzmann model for fluid dynamics. The degree of randomness in the local propagation and collision steps the lattice Boltzmann method (LBM) model controls the turbulent behaviour of the global system, as is illustrated in Fig. This the parameter n that we estimate in our experiments. The results are shown in Fig. and Fig. 9, respectively. Using importance sampling, the topological loss outperforms
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(a) low turbulence
(b) medium turbulence
(c) high turbulence
Figure 7: Left to right: simulations of the fluid model after 3000 iterations, for rj = 0.2, 0.3 and 0.4, respectively. For lower values of y the fluid flow is less turbulent than for higher values of 77-
0.35
0.4
4
4
0.22
0.35
0.25
2
0.18
0.3
0.2
0.16
0.15
0.25
x 3000
3000
3000
3000
3000
3000
(a) Importance sampling
(b) MCMC sampling
Figure 8: Summary of parameter estimates for the parameter of the fluid model, using 5 repetitions of the inference procedure. Each subplot depicts estimates for 17 E {0.20, 0.30, 0.40). respectively. The true parameter is shown as a solid black line. Estimates based on the geometrical loss fg are less accurate than the topological loss fT.
the geometrical loss in terms of accuracy, boxplots in Fig. 8. Moreover, the topological loss in combination with importance sampling also outperforms the geometrical loss with MCMC sampling, evidenced from Table 1. The topological loss in combination with MCMC sampling, however, is far off in terms of accuracy. The latter indicates that the convergence rates of MCMC sampling and importance sampling may be very different, and depend on the given setting. Finally, we note that rejection sampling with both the mean and standard deviation summary statistic does not perform reliably, and its accuracy highly depends on the true paramcter the underlying obscrvation.
be seen from the median and interquartile range of the
Multi-scale structures in percolation models. Fig. 10 shows illustration of our proposed percolation model. The higher the value of the more likely it is for pixel to be non-zero. Consequently, there are more non-zero pixels for higher values of p. As discussed in the main paper, the topological distance outperforms the SCC distance significantly, with both importance sampling and MCMC sampling. However, the rejection sampling procedure together with the mean statistic performs even more accurate in this experiment, as can seen by comparing Fig. and Fig. 12. This not surprising: once the pixel size converges to infinity, the mean of the pixel values will converge the expected pixel value, where the latter only depends on p (since the maximum greyscale value fixed in our experiments, and therefore does not have any discriminative
0.22
0.34
0.32
0.4
0.21
0.3
0.2
0.28
0.35
0.26
0.19
Rejection sampling
Figure Summary parameter estimates for the parameter 7, of the fluid dynamics model with respect summary statistics and a standard rejection sampling procodure, using repetitions of the inference procedure. Each subplot depicts estimates for 17 E (0.20, 0.30, 0.40), respectively. The true parameter is shown solid black line. Estimates based the mean statistic Lzg loss and the standard deviation statistic L2 loss tend to be less reliable than estimates based on the topological loss, see Fig. 8.
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Figure 10: Teft to right: samples the proposed percolation model for p 0.15, 0.3 and 0.6, respectively. All samples admit a fixed maximum greyscale value of 50. Although the geometric distance between two samples of fixed p can be large due the uniform sampling of pixcls, the overall topological structure (which is captured by persistent homology) for given p is more stable with respect. to sampling.
0.62
0.61
1.5
0.18
0.6
4
2
0.16
0.59
2
0.5
0.29
0.58
50
Umax
50
Umna
50
50
50
50
(a) Importance sampling
(b) MCMC sampling
Figure 11: Summary of parameter estimates for the parameter p of the Percolation model, using repetitions of the inference procedure. Each subplot depicts estimates for p {0.15, 0.30, 0.60), respectively. The true parameter is shown solid black line. Estimates based on the geometrical loss Lg less accurate than the topological loss fr, in particular for larger number of iterations.
power for inference). Therefore, at least a largo-pixel regime we can infer the true paramcter of the observation from the mean of its pixel values. Note that this due to the additional structure in this experiment, and does not hold for complex systems that cannot be described in such a simple way, has been seen in the previous experiments. Finally, rejection sampling with the standard deviation statistic leads to highly erroneous results, as can also be seen from Fig. 12. For complex systems it is therefore very difficult to determine the 'right' summary statistic which contains a sufficient amount of information to infer the underlying parameter. This choice is not necessary when using topological losses, which makes the latter an appropriate generic choice, in many applications.
0.33
0.16
0.32
0.64
0.62
0.15
0.3
0.6
0.15
50
50
50
(a) Rejection sampling
Figure 12: Summary of parameter estimates for the parameter p of the Percolation model with respect to summary statistics and a standard rejection sampling procodure, using 5 repetitions of the inference procedure. Each subplot depicts estimates for p E (0.15, 0.30. 0.60), respectively. The true parameter shown solid black line. Estimates based the mean statistic Lag loss statistic L2 loss.
accurate than the standard deviation
19

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Published in Transactions on Machine Learning Research (11/2024)
Table 5: For 500 simulation points: Estimating the randomness parameter of the lattice Boltzmann model after 3000 simulation iterations, using repetitions of the inference procodure. The topological loss together with importance sampling outperforms all of the other configurations.
Randomness parameter
Loss
Sampling method
0.20
0.30
0.40
Importance sampling
0.14 + 0.02
0.22 + 0.02
0.26 + 0.03
la
MCMC
0.18 + 0.07
0.21 + 0.03
0.22 + 0.07
Importance sampling
0.21 + 0.01
0.28 + 0.01
0.38 + 0.05
er
MCMC
2.78 + 1.82
2.91 + 2.11
2.35 + 0.80
Ablation study
Further details and proof
Algorithm MCMC approximation of the target
Input: Observed data y, start value 8, proposal distribution q(0 (0)
Sample Xo p( 0b)
3: for = 1 K do
4: Sample U ~ Unif(0, 1]
5: Sample 8' ~ q(- 1;-1)
Sample X' p(r
Compute
a(0,-1,0') e ((())
((')) 9(0,_1 (')
e-((x.Xi-1)p(0.... 80 11:1)
if U < a(0i-1, 0²) then
Set 0; = 0 X; X
10:
else
11:
Set 0; = 8 1, X; = X;-1
12:
end
13:
end for
14: Output:
0 (01.k).
Description of Algorithm 2
Inputs:
Observed Data y: The actual data you have collected.
Start Value 0o: An initial guess for the parameter 0, whose posterior distribution one intends to estimate.
Proposal Distribution q( (): A distribution used to propose new values for the parameter 0, based on its current value.
Steps:
Initialize: Start by sampling Xo from the distribution p(a (0), where p(x ()) represents the likelihood of the data given parameter 0. This initializes the first sample of the latent variable Xo-
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Published in Transactions on Machine Learning Research (11/2024)
Iterative Sampling (Loop from i ==1 1 to K): For each iteration, repeat the following steps to generate sequence of samples that approximate the distribution of 0:
Generate Random Number U: Sample U ~ Unif(0, 1]. This will be used for the accept- ance/rejection step.
Propose a New Parameter Value 8:: Sample a new candidate value &' from the proposal distribution g( (0:-)), which suggests a new possible value for the parameter based on the current. value 0,-1-
Sample New Latent Variable X':
Sample new value for the latent variable X' from the distributtin p( (0), corresponding to the newly proposed parameter 0'.
Compute the Acceptance Ratio a(0,-1,0); 0)): Calculate the acceptance ratio, which compares the likelihood of the data given the new parameter 6' and the current parameter 0;-1, adjusted by their prior probabilitics and proposal distributions. This ratio helps decide whether the new proposed value 0' should be accepted or not.
Acceptance/Rejeee Rejeection Step: Compare U (the random number sampled earlier) with the acceptance ratio a(0;-1, 8'):
U < a(0,-1,0) 0', accept the new proposed parameter 0' and set 0, 0', and also update the latent variable to X; XX.
Otherwise, reject the proposed value and retain the current parameter and latent variable, iee, set 0, = 0 and X, = X;-1.
Repeat: Continue this process for a total of K iterations.
Output: After completing the iterations, the algorithm outputs sequence sampled parameter values 00,,0, which approximates the posterior distribution of the parameter given the observed data.
Proof about the comparison-based posterior Proposition B.1. The comparison-based posterior
cxp (xp(-((),x)) p(a 0)p(0)
T00 x (v)
Sx exp((((y, p(x p())))(()0d)))) ()p(O)dddr
is the solution (0, x(0, x 11) to the optimisation problem
q² = arg mogmiingeeees (P((xx) [E,[((y,)) + KL (()(((()))))))))())
Proof.
(B = arg min,
(Q) q(x,0) 0)
((y,, ((y,2))(z, () + |flog p(x(0)=(()) {2 0ddrd
arg min,
[// log (exp (exp x) q(x, 0)dxd0 +
// log g(z, 0)
= arg min,
q(x 0)drd0
p(=10)=(0)
{ʃʃ log
g 0)
arg min,
((., 0)drdd
p(x10)) exp (- Cly. x))
[ʃ log
g(x 8)
q(2, 0)dzdd }
arg min,
log Z
p(x)0)=((0) exp -(y, x)) Z-1
arg min, KL (g(x, ()) exp (-((y.x))) 1-1)
21

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Published in Transactions on Machine Learning Research (11/2024)
Hence, the minimiser q is such that
2(2, 0) p(r ()) exp (xpppe((a (-((,,x)) q(0) = f q(2,, f)dr
Z
and
Z = /| exp((e(y,x)) p(z (0)ddddd
Homology in a nutshell Homology groups are a fundamental tool in algebraic topology, providing algebraic representations of the topological structure of a space. Given topological space K, the homology group Ha(K) captures the d-dimensional topological features of the space, such as connected components, loops, and voids. At a high level, the homology group Ha(K) encodes the d-dimensional 'holes' of the space. Ho(K) represents the number of connected components (O-dimensional holes). (I()) captures loops or I-dimensional holes, such as circular paths that do not bound any region. Higher-dimeensonal homology groups H)( with 2 detect higher-dimennionall voids, such as cavities inside the space. Homology groups are defined through an algebraic process. The space K is decomposed into simpler components, such as simplices in the case of simplicial complexes. Chains are formed as formal combinations of these components, and boundary operators are introduced to map d-dimensional simplex to its (d 1)-dimensional boundary. The homology group Ha(K) is then given by the quotient:
Ha(K)
(000a)/imm@0+1)))
where 02 denotes the boundary operator at dimension d. Intuitively, elements of Ha(K) are equivalence classes of d-dimensional cycles (chains with no boundary) in K that are not boundaries of higher-dimensional objects. In summary, the homology group Ha(K,) encodes critical information about the d-dimensional features of a space K, allowing to analyze and compare spaces using algebraic methods.
marginl MCMC in a nutshell Pseudo Pseudo-marginal Markov Chain Monte Carlo (MCMC) methods are a class of algorithms used for sampling from a postcrior distribution when the likelihood function is intractable or expensive compule. In such scenarios, the exact likelihood is replaced by an unbiased estimator, which introduces randomness into the acceptance ratio of the Metropolis SestrooolssHHastings Hastings (MH) algorithm. Despite this randomness, sgeudo-marginal MCMC methods can still produce samples from the correct target distribution. In Bayesian inference, one often needs to sample from a posterior distribution x(0 y) x x (())p(y) (), where x (0) is the prior, and p(y (0) is the likelihood function. In many practical cases, such as when dealing with complex models, the likelihood p(y 0) may not be analytically tractable or computationally feasible to evaluate directly. margin-aar MCMC provides a solution by replacing the likelihood with an unbiased estimator pvy 0). This approach retains the correct target distribution =( y) because the pseudo nreudo-marginal method treats the noisy estimate p(y ()) as if it were the exact likelihood within the Metropolis-Hastingg framework. The pseudo-marginal MCMC algorithu proceeds similarly to the standard Metropolis-Hasti algorithm, with the key difference being that the likelihood ratio in the acceptance probability is replaced by a ratio of likelihood estimators. Specifically, given a current state O,, a new state 0' is proposed, and the acceptance probability is calculated as:
a(1,,0') min (1, ply (z) (0 10)
"ly (0. (0(" 0.))
where p(y (8)) and ply (.) are unbiased estimates of the likelihood, and q (b:) is the proposal distribution. Crucially, the unbiasedness of the likelihood estimator ensures that, despite the randomness introduced, the chain remains reversible and converges to the correct posterior distribution =(0 yy). See Andrieu & Roberts MCMC.
(2009) for thorough introduction to
22


Reviews:
[{"review_id": "bHnne5dfR9", "reviewer": "Reviewer_KcrH", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "**Strengths**\n- The paper is very well-written, with barely any grammatical errors.\n- Good references and theoretical comparisons to related work.\n\n**Weaknesses**\n- The definition of the problem and contribution can be more clear, especially the specific challenge addressed and the benefits brought by tological loss functions.\n- The application scenario can be more clear. The paper mentions various scenarios (intractable/non-differentiable likelihoods, unknown/\"black-box\" likelihoods, complex systems, etc.). However, a precise definition seems missing.\n- Comparison method is relatively limited.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This work lies at the intersection of Bayesian inference and topology. The authors propose a new generalized Bayesian inference approach that uses topological loss functions in parameter estimation. The topological loss can accommodate different data modalities and result in more accurate parameter inference. The authors demonstrate the effectiveness of their approach on three example problems, including the Vicsek model, Lattice Boltzmann Simulation and Percolation Model, and showed their method is more accurate than the geometrical loss function.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "- In Prop. 3.1., $\\pi$ seems abused. Both $\\pi(\\theta, x \\mid y)$ and $\\pi(\\theta)$ are used. The first represents the optimal solution. The second is used in the objective function. What does $\\pi$ denote exactly?\n\n- I am a bit confused by the challenges the authors want to address with their method, which combines topology and Bayesian inference. Initially, my understanding of the targeting problem is cases when the likelihood function is not analytically tractable or differentiable. However, it seems this has already been addressed by generalized Bayesian methods. So what is the exact benefit of introducing topological loss functions? And what is the main challenge that the authors are trying to address?\n\n- Furthermore, the application scenario of the proposed method is not clear. In Introduction and Method sections, the authors mentioned analytically intractable or non-differentiable likelihoods. In conclusion part, the authors mentioned unknown or \"black-box\" likelihood. Additionally, the authors also mentioned \"complex systems\". In contrary to all these, the problems in experiments seem to have known likelihood functions that can be expressed explicitly, thus tractable. So I am a little confused. What is the exact application scenario of the proposed method?\n\n- In Vicsek model, why is the likelihood not analytically tractable? Based on my understanding, the distribution of the position of each agent can be written in closed form at any time step.\n\n- Can the authors clarify in the experiments section whether the likelihood is intractable or non-differentiable for Lattice Boltzmann Simulation and Percolation Model? Why is it necessary or beneficial to use topological loss in these cases?\n\n- I am a bit confused by Section 3.2. \n  - \"In the ensuing discussion, we will illustrate that a modification of the aforementioned approach, specifically,\" Does the aforementioned approach refer to the generalized Bayesian method in Eq. (1)?\n  - \"This approach, however, encounters several obstacles.\" What does \"this approach\" refer to? Is it a naive, intermediate approach proposed by the authors that inspired the final approach? Or is it an existing approach in the literature?\n  - The motivation of Prop. 3.1 is a bit unclear. Is Prop. 3.1 a new form of generalized Bayesian posterior? How is it related to the overarching goal of incorporating topolocial information into Bayesian inference?\n\n- In the experiments, the authors only compared with the geometrical loss function. Is any of the methods mentioned in related works applicable to compare?\n\n- In Table 2, how did the authors define \"outperform\"? When p=0.3, geometrical loss with importance sampling achieved higher accuracy but much larger variance than topological loss ($0.30 \\pm 0.47$ vs. $0.29 \\pm 0.00$). The authors may want to clarify their consideration here, how they determine which is better.", "broader_impact_concerns": "N/A", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "HodNYqAcYi", "reviewer": "Reviewer_WZUE", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n\n1. The paper presents a theoretically grounded method for integrating topological information into Bayesian inference, particularly useful for complex models where traditional likelihood-based approaches are intractable.\n\n2. It demonstrates enhanced performance over geometric losses in multiple simulation studies, highlighting the potential of topology-based losses for accurate parameter estimation.\n\nWeaknesses:\n\n1. The novelty of the paper is a concern, as it primarily applies existing comparison-based posterior approaches with topological losses, rather than introducing fundamentally new concepts.\n\n2. The paper lacks baselines and does not include comparisons to alternative methods for parameter estimation within these settings.\n\n3. The motivation and justification for not using standard gradient-based approaches are unclear, leaving the reader questioning the necessity of the proposed method.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper proposes a novel approach for performing Bayesian inference using topology-based loss functions. The authors develop a framework to leverage topological information in parameter estimation through a comparison-based posterior approach. They introduce topology-based loss functions that can be used within this Bayesian framework and demonstrate how sampling algorithms can utilize these topologically-informed loss functions for Bayesian inference.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "1. Provide a detailed explanation of what distinguishes this work from existing approaches. Highlight the novelty of the methodology beyond the application of topological losses in the comparison-based posterior framework.\n\n2. Include comparisons to relevant baseline methods to demonstrate the added value of the proposed approach. Additionally, perform a more comprehensive analysis of the uncertainty quantification capabilities, including comparisons with traditional methods.\n\n3. Expand on the reasoning behind why standard gradient-based methods cannot be applied to the specific models discussed. This should include a more thorough discussion of the limitations of traditional approaches in these settings.\n\n4. Include a discussion of alternative methods for parameter estimation in complex models, positioning the proposed approach within the broader context of existing research. This will help to better frame the contribution of the paper.\n\n5. Improve the clarity of technical explanations, particularly in sections that may be challenging for readers unfamiliar with topology or Bayesian inference, such as the discussion on pseudo-marginal MCMC.", "broader_impact_concerns": "The paper does not raise any significant ethical concerns.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "1lUjFqJkhJ", "reviewer": "Reviewer_m291", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "**Strengths**:\n- The paper is overall well written, easy to understand.\n- The use of topological-based loss function in the generalized Bayesian inference is of high interest for complex physics systems, to build an accurate posterior distribution. \n- The explanation of the topological tools is accessible for non-expert readers.\n- The choice of the topological complexes in each numerical setting is well justified.\n\n**Weaknesses**:\n- The numerical experiments are limited to very-low dimensional settings (only 1D for complex systems, 2D for synthetic data). It is unclear to state if the proposed method may scale to higher dimensional settings under reasonable amount of simulations in the sampling phase (in particular, I am quite concerned about the IS approach which is known to scale poorly with dimension). Note that this is not indicated as a limitation of the method.\n- The overall computational budget of the method is unclear to me: in particular, the topological based approach requires to compute a Wasserstein-2 distance which is known to scale very poorly with respect to the data dimension and to the number of points. To ensure fair comparison with the geometry-based approach, the cost of the loss computation should also appear.\n- The dependence to hyperparameters is unclear: especially, about the number of scales in the topological part and the design of the Markov kernel on $\\theta$ in the MCMC setting (which is highly crucial).\n- It is not clear why the exact Wasserstein distance is preferred to the entropy-regularized Wasserstein distance (which is known to be more efficient in practice).", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "In this paper, the authors present a methodology to perform Bayesian inference (i.e., sampling from a posterior distribution) with data topological knowledge, which is a setting that is still largely unexplored in model parameter estimation. This approach is specifically designed for scenarios where the likelihood of the statistical model is only available via simulations (simulation-based inference). In practice, these simulations are given by a generative model; here, the likelihood of this model is also assumed to be non-differentiable with respect to the parameter, which makes impossible to use gradient-based approaches such as variational inference.\n\nThe proposed method builds on the framework of generalized Bayesian inference by introducing a topological-based loss function in the construction of a two-state posterior distribution (combining model parameter and simulated data). This loss is computed as the Wasserstein-2 distance between sets of multi-scale topological features obtained from real data and simulated data (namely, Victoris-Rips complexes and cubical complexes, depending on the application). This consists in the main contribution of the paper. Once the generalized posterior is built, Bayesian inference is conducted by directly sampling from this distribution either via importance sampling or Metropolis-Hastings MCMC method. This approach is supported by numerical experiments conducted on complex physics systems, which compare the use of topological-based loss vs geometry-based loss under the same budget of simulations. In all of the presented settings, the topological-based approach is on par or outperforms the geometry-based approach.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "**Critical**:\n- In my opinion, **an ablation study on the number of simulations** in each numerical setting is missing. This study should inform how robust the proposed method is with respect to the sampling complexity, and how well it performs when increasing this budget. For instance, plotting the parameter estimation error vs the sampling complexity could be a good idea.\n- The **limitation to low dimensional cases** (here, I am referring to the parameter dimension) should be addressed either by performing new numerical experiments in higher dimension or clearly stating this limitation in the paper.\n- The **hyperparameters of the method should be clearly stated** in each numerical experiment (see the 'Weaknesses' section), and tested with small ablation studies to validate robustness. For instance, with good tuning of the Markov kernel in the MCMC algorithm, I am quite convinced that the MCMC sampling could be as good as importance sampling in each setting.\n- More details on **the computation of the Wasserstein-2 distance** should be given: complexity in practice (which is not trivial at all), preference over regularized Wasserstein-2 distance.\n\n**Moderate**:\n- The field name 'Simulation-based inference' with related work should appear in the introduction, as it encompasses the scenario chosen by the authors for Bayesian inference.\n- The Algorithm 2 should be explained by sentences which give the main infos, i.e., computing a MH rate on a two-state distribution, making explicit the two-state proposal Markov kernel (give advantages and drawbacks), explaining why the MH rate simplifies in $x$... As such, it is not commentated at all, which may be hard to understand for non-expert readers.", "broader_impact_concerns": "None.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Meta-review: The authors have already made revisions to the manuscript in response to the reviewer comments and have engaged in a robust discussion with the reviewers about the manuscript. The manuscript is much improved as a result of the revisions to the reviewer comments.
Decision: Accept (poster)