Spectral geometric methods in practice

Projektinformationen

SPECGEO

ID Finanzhilfevereinbarung: 802554

DOI

10.3030/802554

Projekt abgeschlossen

EK-Unterschriftsdatum 31 August 2018

Startdatum 1 September 2018

Enddatum 31 August 2024

Finanziert unter

• EXCELLENT SCIENCE - European Research Council (ERC)

Gesamtkosten

€ 1 434 000,00

EU-Beitrag

€ 1 434 000,00

Koordiniert durch

UNIVERSITA DEGLI STUDI DI ROMA LA SAPIENZA
Italy

Periodic Reporting for period 4 - SPECGEO (Spectral geometric methods in practice)

Berichtszeitraum: 2023-03-01 bis 2024-08-31

Zusammenfassung vom Kontext und den Gesamtzielen des Projekts

Geometric data, encompassing shape, connectivity, and proximity, are ubiquitous: Our three-dimensional bodies and surroundings, the complex structure of proteins carrying critical biological information, the human connectome's functional and anatomical connectivity, and social networks modeling relationships as graphs are just a few examples. Similarly, deep learning architectures, sensor data, and dynamic networks underpin key tasks in computer vision and beyond.

The exponential growth of digital data collected through computers and sensors has transformed nearly every aspect of life. However, in this era of "big data," reliance on manual supervision is no longer feasible. Artifacts like missing and corrupted data are commonplace, posing significant challenges. Robust methodologies are essential to extract meaningful insights from such imperfect data. The application of spectral techniques to high-dimensional geometric data undergoing real-world transformations, including deformation, incompleteness, and noise, remains limited. These challenges require a paradigm shift to develop methods that are both resilient to corrupted inputs and capable of delivering reliable results. This project sought to create a new generation of spectral methods for analyzing and processing geometric data under complex transformations and corruption.

Our ultimate goal was to design a theoretical spectral framework and computational methods for geometric data analysis, addressing issues like topology changes, noise, and cross-modality representation. Research targeted fields where geometric data are pivotal, including computer vision, graphics, machine learning, and computational biology, while also impacting areas like acoustics, mechanical engineering, and computational sociology.
The project surpassed its objectives, delivering innovative spectral geometric methods and computational frameworks to address challenges in geometric data processing. These approaches are robust, versatile, and impactful across diverse domains.
A particularly exciting outcome was the synergy between our methods and machine learning. The spectral geometric techniques developed extended beyond their original focus to enable advances in graph-based and representation learning, significantly increasing the project's reach and visibility. This alignment with the rapid growth of machine learning spurred collaborations and practical applications.

The project has solidified the PI's research group's standing at the intersection of machine learning, computer vision, and geometry processing. Previously centered on geometry and vision, the group now plays a prominent role in the machine learning community, with impactful publications and contributions that highlight its broader relevance.

Arbeit, die ab Beginn des Projekts bis zum Ende des durch den Bericht erfassten Berichtszeitraums geleistet wurde, und die wichtigsten bis dahin erzielten Ergebnisse

A major contribution of the project was the development of shape-from-spectrum techniques to address inverse problems. Initially a sub-goal, these methods became a cornerstone of the project and were enthusiastically received by the research community. This success led to unexpected applications, such as the discovery of spectral adversarial attacks on geometric data like deformable 3D shapes, laying the groundwork for further research in this area.

On the theoretical front, we made significant strides in understanding eigenspace dynamics and the effects of geometric perturbations (e.g. missing data), publishing these results in leading venues for digital geometry processing. Inverse spectral geometric problems became a central focus of our work, garnering notable recognition, including awards. We advanced both axiomatic (learning-free) and data-driven (learning-based) approaches to tackle these challenges, including the famous "hearing the shape of the drum" problem. Maps and correspondence problems provided an ideal setting for showcasing our computational techniques, while the spectral representation of maps on product manifolds was thoroughly explored and published early in the project.

The computational framework proposed at the outset was successfully developed and released as a comprehensive open-source codebase, enriched with functionality corresponding to each publication. Accompanied by datasets, the framework ensures reproducibility and serves as a valuable resource for applied data sciences. The project resulted in numerous novel methodologies, documented in several dozen publications at top-tier conferences and journals spanning computer vision, machine learning, computer graphics, and geometry processing, many with open-source implementations.

Dissemination was another key success of the project. Our team delivered numerous invited talks, keynotes, panels, and seminars at major international institutions and conferences. We also organized widely attended workshops and tutorials, fostering collaboration and knowledge exchange in both scientific and industrial communities. Many of the project's methods are now state-of-the-art and serve as benchmarks for evaluating new approaches. By advancing spectral geometric methods and extending their reach into machine learning—initially a secondary goal but ultimately a major achievement—the project exceeded its objectives, leaving a lasting impact across multiple fields.

Fortschritte, die über den aktuellen Stand der Technik hinausgehen und voraussichtliche potenzielle Auswirkungen (einschließlich der bis dato erzielten sozioökonomischen Auswirkungen und weiter gefassten gesellschaftlichen Auswirkungen des Projekts)

Spectral methods had a lasting impact on applied sciences, delivering unprecedented performance and novel capabilities across fields involving geometric data. By operating at a fundamental level rather than offering ad-hoc solutions for specific tasks, our methodology proved applicable to a wide range of open problems in various domains. It also demonstrated potential to address challenges that were previously unimaginable but are likely to emerge in the future. The remarkable growth of geometric data across scientific domains in recent years made this research particularly timely.

The tangible benefits of our work were evident in computer vision and graphics, fields where geometric data has been a central focus since their inception. Our spectral methods advanced the state of the art in shape correspondence problems, leveraging both axiomatic and data-driven approaches. Notably, these methods addressed persistent challenges such as geometry artifacts, partial data, and deviations from isometry. This progress helped to overcome skepticism within the geometry processing community, which had been discouraged by the perceived instability of spectral constructions in classical theory.

Unexpectedly, we also achieved significant success in addressing inverse spectral geometric problems ("hearing the shape of the drum") and adversarial attacks on geometric data. The former resulted in three top-tier publications, including two awarded Best Paper prizes at international conferences, marking this as a highly promising research avenue. The latter, focusing on spectral adversarial perturbations, proved less challenging than anticipated, culminating in two major publications. These successes were partly enabled by advances in our work on inverse problems, as the flexible methodologies developed there served as effective computational tools for adversarial scenarios.

Overall, the project exceeded expectations, demonstrating the versatility and impact of spectral methods across a broad spectrum of applications while opening new directions for future research.

Spectral geometry processing