A.1 Cellular WLResults In this section, we assume basic familiarity with the WL test and its higher-order variants. For an introduction to these topics, we refer the reader to the survey of Sato [62]. We begin by introducing a few useful concepts. A cellular colouring is a map c that maps a cell complex X and one of its cells to a colour from a fixed colour palette. Let X,Y be two regular cell complexes and c a cellular colouring. We say that X,Y are c-similar, denoted by cX = cY, if the number of cells in X coloured with a given colour equals the number of cells in Y with the same colour. Otherwise, we have cX 6= cY . We emphasise that in this paper we are interested only in colourings c with the property that any two isomorphic cell complexes are c-similar. A cellular colouring c refines a cellular colouring d, denoted by c v d, if for all cell complexes X and Y and all 2 PX and 2 PY, cX = cY implies dX = dY . Additionally, if d v c, we say the two colourings are equivalent and we represent it by c d. We state the following result from Bodnar et al. [8] about simplicial colourings, which we translate here directly to cell complexes. The proof is however, identical, and we refer the reader to their work for that. Let X,Y be any regular cellular complexes with A PX and B PY . Consider two cellular colourings c,d such that c v d.

In this work, we investigate the geometry of the k-th homology embedding and focus on cases reminiscent of spectral clustering. Namely, we analyze the connected sum of manifolds as a perturbation of the direct sum of their homology embeddings.

We develop a framework for comparing data manifolds, aimed, in particular, towards the evaluation of deep generative models.

Equivalently,ifwe demonstrate that anylinear function obtains the same maximum value overS asoverCH,theproofiscomplete. Figure 1inourmain paper compares thetightness between different relaxations. For using the Anderson relaxation, we need to replace the input simplex with the unit hypercube. The relaxation first uses Kuhn triangulation of[0,1]n [Todd, 1976], which is used to describe the collection of simplices whose union is[0,1]n. Then constraintsr1 andr2 are constructed using these two triangles. In contrast, our relaxation works directly with the input simplex and only requires one upper constraint.

It is increasingly common for data to possess intricate structure, necessitating new models and analytical tools. Graphs, a prominent type of structure, can encode the relationships between any two entities (nodes). However, graphs neither allow connections that are not dyadic nor permit relationships between sets of nodes. We thus turn to simplicial complexes for connecting more than two nodes as well as modeling relationships between simplices, such as edges and triangles. Our data then consist of signals lying on topological spaces, represented by simplicial complexes. Much recent work explores these topological signals, albeit primarily through deterministic formulations. We propose a probabilistic framework for random signals defined on simplicial complexes. Specifically, we generalize the classical notion of stationarity. By spectral dualities of Hodge and Dirac theory, we define stationary topological signals as the outputs of topological filters given white noise. This definition naturally extends desirable properties of stationarity that hold for both time-series and graph signals. Crucially, we properly define topological power spectral density (PSD) through a clear spectral characterization. We then discuss the advantages of topological stationarity due to spectral properties via the PSD. In addition, we empirically demonstrate the practicality of these benefits through multiple synthetic and real-world simulations.