arXiv subjects

Sara Scaramuccia

Publications and source records attributed to Sara Scaramuccia.

5 recordsLinked to original sources

arXiv2026

The Convex Matching Distance in Multiparameter Persistence

We introduce the convex matching distance, a novel metric for comparing functions with values in the real plane. This metric measures the maximal bottleneck distance between the persistence diagrams associated with the convex combinations of the two function components. Similarly to the traditional matching distance, the convex matching distance aggregates the information provided by two real-valued components. However, whereas the matching distance depends on two parameters, the convex matching distance depends on only one, offering improved computational efficiency. We further show that the convex matching distance can be more discriminative than the traditional matching distance in certain cases, although the two metrics are generally not comparable. Moreover, we prove that the convex matching distance is stable and characterize the coefficients of the convex combination at which it is attained. Finally, we demonstrate that this new aggregation framework benefits from the computational advantages provided by the Pareto grid, a collection of curves in the plane whose points lie in the image of the Pareto critical set associated with functions assuming values on the real plane. Experimental validation on MNIST digits, synthetic shapes, and chaotic attractors suggests that the convex matching distance provides a reliable and efficient alternative to the matching distance, at a significantly lower computational cost.

math.AT↗

arXiv2025

Parallel computation of interval bases for persistence module decomposition

A persistence module $M$, with coefficients in a field $\mathbb{F}$, is a finite-dimensional linear representation of an equioriented quiver of type $A_n$ or, equivalently, a graded module over the ring of polynomials $\mathbb{F}[x]$. It is well-known that $M$ can be written as the direct sum of indecomposable representations or as the direct sum of cyclic submodules generated by homogeneous elements. An interval basis for $M$ is a set of homogeneous elements of $M$ such that the sum of the cyclic submodules of $M$ generated by them is direct and equal to $M$. We introduce a novel algorithm to compute an interval basis for $M$. Based on a flag of kernels of the structure maps, our algorithm is suitable for parallel or distributed computation and does not rely on a presentation of $M$. This algorithm outperforms the approach via the presentation matrix and Smith Normal Form. We specialize our parallel approach to persistent homology modules, and we close by applying the proposed algorithm to tracking harmonics via Hodge decomposition.

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arXiv2021

Relative-perfectness of discrete gradient vector fields and multi-parameter persistent homology

The combination of persistent homology and discrete Morse theory has proven very effective in visualizing and analyzing big and heterogeneous data. Indeed, topology provides computable and coarse summaries of data independently from specific coordinate systems and does so robustly to noise. Moreover, the geometric content of a discrete gradient vector field is very useful for visualization purposes. The specific case of multivariate data still demands for further investigations, on the one hand, for computational reasons, it is important to reduce the necessary amount of data to be processed. On the other hand, for analysis reasons, the multivariate case requires the detection and interpretation of the possible interdepedance among data components. To this end, in this paper we introduce and study a notion of perfectness for discrete gradient vector fields with respect to multi-parameter persistent homology, called relative-perfectness. As a natural generalization of usual perfectness in Morse theory for homology, relative-perfectness entails having the least number of critical cells relevant for multi-parameter persistence. As a first contribution, we support our definition of relative-perfectness by generalizing Morse inequalities to the filtration structure where homology groups involved are relative with respect to subsequent sublevel sets. In order to allow for an interpretation of critical cells in $2$-parameter persistence, our second contribution consists of two inequalities bounding Betti tables of persistence modules from above and below, via the number of critical cells. Our last result is the proof that existing algorithms based on local homotopy expansions allow for efficient computability over simplicial complexes up to dimension $2$.

cs.CG↗

arXiv2020

Feedback Clustering for Online Travel Agencies Searches: a Case Study

Understanding choices performed by online customers is a growing need in the travel industry. In many practical situations, the only available information is the flight search query performed by the customer with no additional profile knowledge. In general, customer flight bookings are driven by prices, duration, number of connections, and so on. However, not all customers might assign the same importance to each of those criteria. Here comes the need of grouping together all flight searches performed by the same kind of customer, that is having the same booking criteria. The effectiveness of some set of recommendations, for a single cluster, can be measured in terms of the number of bookings historically performed. This effectiveness measure plays the role of a feedback, that is an external knowledge which can be recombined to iteratively obtain a final segmentation. In this paper, we describe our Online Travel Agencies (OTA) flight search use case and highlight its specific features. We address the flight search segmentation problem motivated above by proposing a novel algorithm called Split-or-Merge (S/M). This algorithm is a variation of the Split-Merge-Evolve (SME) method. The SME method has already been introduced in the community as an iterative process updating a clustering given by the K-means algorithm by splitting and merging clusters subject to feedback independent evaluations. No previous application of the SME method to the real-word data is reported in literature to the best of our knowledge. Here, we provide experimental evaluations over real-world data to the SME and the S/M methods. The impact on our domain-specific metrics obtained under the SME and the S/M methods suggests that feedback clustering techniques can be very promising in the handling of the domain of OTA flight searches.

cs.IR↗

arXiv2018

Computing multiparameter persistent homology through a discrete Morse-based approach

Persistent Homology (PH) allows tracking homology features like loops, holes and their higher-dimensional analogs, along with a single-parameter family of nested spaces. Currently, computing descriptors for complex data characterized by multiple functions is becoming an important task in several applications, including physics, chemistry, medicine, geography, etc. Multiparameter Persistent Homology (MPH) generalizes persistent homology opening to the exploration and analysis of shapes endowed with multiple filtering functions. Still, computational constraints prevent MPH to be feasible over real-sized data. In this paper, we consider discrete Morse Theory as a tool to simplify the computation of MPH on a multiparameter dataset. We propose a new algorithm, well suited for parallel and distributed implementations and we provide the first evaluation of the impact on MPH computations of a preprocessing approach.

cs.CG↗