In this work, we consider the problem of intersectional group fairness in the classification setting, where the objective is to learn discrimination-free models in the presence of several intersecting sensitive groups.
First, we illustrate various shortcomings of existing fairness measures commonly used to capture intersectional fairness. 
Then, we propose a new definition called the $\alpha$-Intersectional Fairness, which combines the absolute and the relative performance across sensitive groups and can be seen as a generalization of the notion of differential fairness. 
We highlight several desirable properties of the proposed definition and analyze its relation to other fairness measures.
Finally, we benchmark multiple popular in-processing fair machine learning approaches using our new fairness definition and show that they do not achieve any improvement over a simple baseline. 
Our results reveal that the increase in fairness measured by previous definitions hides a ``leveling down'' effect, i.e., degrading the best performance over groups rather than improving the worst one.

Fair Without Leveling Down: A New Intersectional Fairness Definition

Topological data analysis (TDA) is a branch of computational mathematics, bridging algebraic topology and data science, that provides compact, noise-robust representations of complex structures. Deep neural networks (DNNs) learn millions of parameters associated with a series of transformations defined by the model architecture resulting in high-dimensional, difficult to interpret internal representations of input data. As DNNs become more ubiquitous across multiple sectors of our society, there is increasing recognition that mathematical methods are needed to aid analysts, researchers, and practitioners in understanding and interpreting how these models' internal representations relate to the final classification. In this paper we apply cutting edge techniques from TDA with the goal of gaining insight towards interpretability of convolutional neural networks used for image classification. We use  two common TDA approaches to explore several methods for modeling hidden layer activations as high-dimensional point clouds, and provide experimental evidence that these point clouds capture valuable structural information about the model's process. First, we demonstrate that a distance metric based on persistent homology can be used to quantify meaningful differences between layers and discuss these distances in the broader context of existing representational similarity metrics for neural network interpretability. Second, we show that a mapper graph can provide semantic insight as to how these models organize hierarchical class knowledge at each layer. These observations demonstrate that TDA is a useful tool to help deep learning practitioners unlock the hidden structures of their models. 

Experimental Observations of the Topology of Convolutional Neural Network Activations

Topological magnetic semimetals, like Co3Sn2S2 and Co2MnGa, display exotic transport properties, such as large intrinsic anomalous (AHE) due to uncompensated Berry curvature. The highly symmetric XPt3 compounds display anti-crossing gapped nodal lines, a driving mechanism in the intrinsic Berry curvature Hall effects. Uniquely, these compounds contain two sets of gapped nodal lines that harmoniously dominate the Berry curvature in this complex multi-band system. We calculate a maximum AHE of 1965 S/cm in the CrPt3 by a state-of-the-art first principle electronic structure. We have grown high-quality thin films by magnetron sputtering and measured a robust AHE of 1750 S/cm for different sputtering growth conditions. Additionally, the cubic films display a hard magnetic axis along [111] direction. The facile and scalable fabrication of these materials makes them ideal candidates for integration into topological devices [1].

![](https://assets.underline.io/uploads/markdown_image/1/image/4135773548df436a10a909902db4a303.jpg)

Hard magnet topological semimetals in XPt3 compounds with the harmony of Berry curvature

A major factor contributing to the success of modern representation learning is the ease of performing various vector operations. Recently, objects with geometric structures (eg. distributions, complex or hyperbolic vectors, or regions such as cones, disks, or boxes) have been explored for their alternative inductive biases and additional representational capacities. In this work, we introduce Box Embeddings, a Python library that enables researchers to easily apply and extend probabilistic box embeddings. Fundamental geometric operations on boxes are implemented in a numerically stable way, as are modern approaches to training boxes which mitigate gradient sparsity. The library is fully open-source, and compatible with both PyTorch and TensorFlow, which allows existing neural network layers to be replaced with or transformed into boxes effortlessly. In this work, we present the implementation details of the fundamental components of the library, and the concepts required to use box representations alongside existing neural network architectures.

Box Embeddings: An open-source library for representation learning using geometric structures

We present a review of geological and seismological data on the long-lived Schio-Vicenza Fault System (SVFS) at the northern Adria plate margin and provide its geodynamic and seismotectonic interpretation. The SVFS is a NW-SE trending, high angle complex structure transverse to the Southern Alps fold-and-thrust belt, and has been the object of geological and structural studies for more than a century and a half. The main segment of the SVFS is the Schio-Vicenza Fault (SVF), which has a significant imprint in the landscape across the Eastern Southern Alps and the Veneto-Friuli foreland. The SVFS is divided into a northern segment, extending across the chain between the town of Schio and the Adige Valley, and a southern one, coinciding with the SVF proper. The SVF borders to the east a foreland structural and morphological high formed by the Lessini and Berici Mts., where the Mesozoic bedrock is exposed, and by the volcanic Euganei Hills, separating it from the Veneto-Friuli foreland, and continuing southeastward up to the Adriatic coastline along the blind Conselve-Pomposa Fault. The segments of the SVFS have been active during several tectonic phases with different styles of faulting at least since the Mesozoic, with a well defined long-term dip-slip component of faulting and, conversely, a horizontal one not well constrained. The SVFS interrupts the continuity of the Southern Alps thrust fronts in the Veneto sector, suggesting that at least it played a passive role in controlling the geometry of the active thrust belt, and possibly also the current distribution of seismic release. In this context, the SVFS, and specifically its SVF segment, has been interpreted as the sinistral strike-slip boundary of the northeastern Adriatic indenter. The recent activity of the SVFS is highlighted by the moderate seismicity along the northern segment and few geological observations along the southern one. The review of the historical and instrumental seismicity along with active deformation data, shows that the SVFS does not appear to have generated large earthquakes during the last few hundred years, while the focal mechanisms of moderate seismicity points to a dextral strike-slip activity, driven by a N-S oriented compressional regime. These observations are corroborated by the field analysis of antithetic Riedel structures of the fault cropping out along the northern segment. Conversely, the southern segment shows geological evidence of recent sinistral strike-slip activity. The geological and seismological apparently conflicting data can be reconciled considering the sinistral strike-slip faulting style of the southern segment as driven by the indentation of the Adriatic plate, while the dextral strike-slip style along the northern segment can be explained as a sinistral opening "zipper" type shear zone junction, where intersecting pairs of simultaneously active faults with different sense of shear merge into a single fault system through a zippered section.

Role of the long-lived Schio-Vicenza Fault System at the northern Adria plate margin

The recent tectonic activity of the Susans Tricesimo Thrust System (STTS) was investigated through a multidisciplinary approach. The study area is located in the northernmost portion of the Friuli eastern prealpine border where the SSE verging front of the Southalpine Chain meets the NW-SE trending strike slip structures of the eastern Friuli-western Slovenia domain (Vrabec & Fodor, 2006; Poli & Zanferrari, 2018). At present, the 2-3 mm/yr accumulating deformation (Serpelloni et al., 2016) is released through the interaction among reverse, transpressive and strike-slip faults. The study area corresponds to the epicentral zone of two of the strongest earthquakes of NE Friuli: the 1511 Mw 6.3 and the 1976-77 (Mw ≤ 6.4) sequence (Rovida et al., 2021). In more recent times, two seismic sequences were registered in western Slovenia in 1998 (Mw 5.6) and 2004 (Mw 5.2). In order to further explore the involvement of the STTS during the strongest seismic events occurred in the area, this work is focused on the recent tectonic activity of the structure. Based on a multidisciplinary approach, the deep 3D-geometry of the STTS was reconstructed by combining ENI seismic lines interpretation and hypocentral distribution analysis. The morphotectonic survey conducted in the study area allowed to detect many surficial anomalies at the hanging-wall of the ST thrust. The collected morphotectonic hints clearly depict a NW-SE elongated zone of surficial deformation, highlighting the recent activity of the Colle Villano-W inner thrust. Lastly, the realization of a paleoseismological trench across the Colle Villano-W thrust revealed that the excavated glacial units of the Canodusso Subsynthem (18.000 - 19.000 yr 14C BP) experienced at last two deformative episodes. Particularly, the occurrence of the latest event in historical times seems likely, and the possible correlation with the 1511 or 1976 events must be considered.

New evidence of recent tectonic activity of the Susans-Tricesimo thrust-system (NE Italy)

The European Marine Observation and Data Network (EMODnet) is a Project aimed at the collection and harmonization in European Seas of existing data, to be represented on digital maps freely available through a dedicated portal (www.emodnet.eu), in order to facilitate their exchange and re-use. It is subdivided into thematic Lots concerning Bathymetry, Geology, Biology, Chemistry, Physics, Seabed Habitats and Human activities. Each Lot is carried out by a consortium of Institutions collating data for its thematic portal also by means of machine to machine communication. The Geology Lot (www.emodnet-geology.eu) is realized by a Consortium of European Geological Surveys. Information includes data at multiple scale on seafloor sediments grainsize, sedimentation rates, Quaternary geology, pre-Quaternary geology and stratigraphy, coastal behaviour, geological events, mineral resources, submerged landscapes. Metadata and available information collected during the Project are displayed on the Portal. The spatial data, organized in layers, are represented by the geometric primitives lines, points and polygons and the alphanumeric data are stored in harmonized and comprehensive databases, which can be consulted online through Attribute tables. Within the Project a subgroup of Geological Surveys of circum-Adriatic countries (Slovenia, Croatia, Montenegro, Albania and Italy) has been established in order to work together and provide already harmonized products to the Project. Elaboration has been carried out concerning pre-Quaternary units, as required by the Project, as well as pre-Pliocene units, considered by common agreement more meaningful in the Adriatic setting. In addition, other features to be represented were taken into account within the task “Geological events and probabilities”, coordinated by the Geological Survey of Italy – ISPRA. They include submarine landslides, earthquakes, volcanic centres, tsunamis, fluid emissions and Quaternary tectonics. However, the best achievement of the Adriatic group was the realization of a map of sediment grainsize at 1:250,000 scale, obtained thanks to the collaboration of CNR-ISMAR as well. Products elaborated for the entire Adriatic Sea highlight the continuity of geological units, as shown by the map of pre-Quaternary deposits, contribute to understand the geological evolution, as is the case of the map of pre-Pliocene deposits, and to outline sedimentary dynamics, as evidenced by the map of seafloor sediments. Moreover, collation of data across national boundaries allows to analyse in their entirety specific features, such as submarine landslides.

Harmonization of geological data in the Adriatic Sea

Orogenic plateaus represent the most spectacular consequence of continental collisions in orogenic belts. The mechanisms related to the growth of an orogenic plateau are still debated and can be the result of multiple tectonic phases and changes in collisional dynamics between the involved plates. In the last decades, the study of landscapes and landforms experienced a significative growth providing new advanced numerical models to relate surface deformation, topographic evolution and plate dynamics. In the Mediterranean area, the Central Anatolian Plateau represents a good example of how different mantle and crustal processes can be involved in building a plateau. The study of the topographic features alongside the plateau margins, such as drainage systems and marine terraces, revealed significantly different uplift histories and mechanisms characterizing the northern and the southern margins. While the northern margin of the plateau experienced vertical uplift from 11Ma ca to the present that can be associated with the transpression along the Northern Anatolian Fault, the southern margin records a fast uplift phase during Pleistocene, causing 1500 km of vertical uplift in about 600.000 years, which has been related to the break-off of the subducting African plate. In our work, we present a summary of how the numerical modeling of the evolution of drainage systems and marine terraces provided significant insights on the uplift history and the topographic evolution of the Central Anatolian Plateau margins, and how these results are linked to the geodynamical processes involving the Anatolian plate boundaries.

Landscape evolution and plate boundary processes: the study case of the Central Anatolian Plateau margins

The Central Mediterranean region in the Tertiary was characterized by intermittent phases of subduction and collision, the opening of back-arc extensional basins (i.e., Liguro-Provençal, Alboran, and Tyrrhenian basins) and episodes of slab lateral tearing, segmentation and break-off that led to the currently complex geological setting. Although the shallow tectonic evolution of the region has been relatively well constrained by abundant geological data, several uncertainties persist about the mechanisms that generated the present-day surface morphology and deep slab geometry. In this study, we reproduce the recent large-scale evolution of the Central Mediterranean and the associated strain-induced upper mantle fabrics and seismic anisotropy combining geodynamic and seismological numerical modelling techniques. 3D petrological-thermo-mechanical numerical models have been carried out using the viscoplastic code I3MG (Gerya, 2019), to compute the LPO and the synthetic SKS splitting we use a modified version of D-REX (Kaminski et al., 2004) and the softwares package FSTRACK (Becker, 2006), respectively. The subduction models were designed and calibrated according to paleogeographic-tectonic reconstructions of 30 Ma and seismological observations from the Mediterranean region available in the literature. We test different initial configurations in order to optimize the fit between predicted and observed slabs position and obtain a final model configuration resembling the present-day surface and deeper structures. It is found that the opening of back-arc extensional basins in response to the south-eastward retreat of the Ionian slab is a common feature in all models and slab lateral tearing or break-off occurs when Adria continental margin enters the trench. More importantly, we show that structural heterogeneities within the Adria plate and/or the geometry of its Tyrrhenian passive margin have a profound impact on (i) the development of a slab window below the Central Apennines, (ii) the retreat of the Northern Apenninic trench till the Adriatic Sea, and (iii) the retreat of the Ionian slab till the present-day position. In general, this study highlights the importance of coupling macro-scale geodynamic modelling with micro-scale simulations of strain-induced upper mantle fabrics and seismological synthetics in order to better constrain the tectonic evolution of complex convergent margins such as the Central Mediterranean. 

Geodynamic and seismological modelling of the recent dynamics of the Central Mediterranean region

The analysis of pressures and impacts on the Apulian aquifers, as required by the EU WFD, has evidenced physical and chemical parameters related to groundwater salinization, particularly along the coastal strip. Predisposing factors such as lithology, tectonic evolution, aquifer over-exploitation (Passarella et al., 2017), and the ongoing climate change are various. Therefore, distinguishing natural from anthropogenic sources of salinization becomes a fundamental issue for assessing the aquifer's chemical status. In this framework, isotopic measures of O, H, B, and Sr, suitably support distinguishing different sources of salinity, water-rock interaction processes, and the origin of the water molecules (Pennisi et al., 2006). This study focuses on the coastal sector of the Murgia aquifer located on the Adriatic side of the Apulia region (south Italy). It is made up of several hundred-meter-thick Mesozoic calcareous and calcareous-dolomite rocks affected by fracturing and karst phenomena. The aquifer is confined and characterized by irregular geometry. Groundwater flows to the sea where it rises in numerous coastal springs. Paleo-seawater as an additional source of salinity has been suggested by previous studies.

A multi-isotope (O, H, B, Sr) approach for identifying salinity contamination along the coastal sector of Murgia aquifer (Apulia, Southern Italy)

Earthquakes can produce a wide spectrum of surface deformations associated with the main events or between inter-seismic periods. The larger events (Mw > 5.5) can trigger deformations such as surface ruptures related to the activation of main active faults and/or other deformations induced by seismic shaking (e.g. landslides, creeping, sink-hole). In the last three decades remote sensing acquisitions (e.g. Synthetic Aperture Radar Interferometry, Lidar differencing, optical imagery and GPS) have reached an impressive detail, useful for detecting high accuracy deformations. In particular, Differential SAR Interferometry (DInSAR), is one of the most powerful and reliable techniques to provide a snapshot of the coseismic deformation generated by earthquakes occurred within complex geological and morphological settings. In 2016-2017, a long earthquake sequence struck the Apennines in central Italy, producing impressive surface ruptures due to the activation of the SW-dipping extensional M. Vettore fault system (VF). Most of these ruptures are attributed to the 24 August Mw 6.0 and 30 October Mw 6.5 main-shocks and have been investigated by several groups of field geologists soon after the earthquakes. The complex distribution of the observed ruptures reflects the complexity of the geological setting of the epicentral area, where a succession of both carbonate and siliciclastic rocks crops out. Here we present detailed maps of the complex deformation pattern produced by the VF during the October 2016 earthquake in central Italy, derived from ALOS-2 SAR data via DInSAR technique. On these maps, we traced a set of sections to analyse the coseismic vertical displacement, necessary to identify both surface fault ruptures and off-fault deformations, the latter not directly recognizable in the field. At a local scale, we identified a large number of surface ruptures at an unprecedented level of details, improving those previously observed in the field. At a regional scale, the deformation pattern depicts a roll-over anticline, formed at the hangingwall of a main activated fault, with a listric geometry.  The rapid detection of deformation patterns from DInSAR technique can provide important constraints on the spatial distribution of the activated fault segments and their interaction. Such information can be crucial for emergency management by civil protection and helpful to drive and support the geological field surveys during the ongoing seismic crisis. These results demonstrate how the proposed approach can be applied for any worldwide earthquake, aimed to provide a wider and faster picture of the surface deformation in areas characterized by complex geology and morphology.

Complex deformation pattern of the October 2016 central Italy earthquakes from DInSAR data

The Friuli Venezia Giulia region (FVG) belongs to the easternmost portion of the Southern Alps, where NE-SW trending, S-SE-verging thrusts of the Southalpine Quaternary front interacts with NW-SE-striking right-lateral strike-slip fault-system that characterizes the Italian-Slovenian border area. Both fault systems are still active as testified by historical and instrumental seismicity (e.g. Rovida et al., 2021). During the last millennium six M≥6 earthquakes struck FVG and surrounding regions causing thousands of casualties and widespread damages (1348, Alpi Giulie; 1511, Friuli-Slovenia; 1928, Carnia; 1873, Alpago; 1936, Cansiglio; 1976, Friuli). Despite the abundance of scientific literature produced after the devastating 1976 Friuli earthquake sequence (see for a review Carulli, 2018), before this project FVG was not equipped with a systematic and homogeneous database of active or potential active faults based on the seismotectonic features. For this reason, the Geological Service of the Friuli Venezia Giulia Region promoted a research project in collaboration with the University of Trieste, the University of Udine and the National Institute of Oceanography and Experimental Geophysics (OGS) of Trieste in order to create a database of the active faults in FVG. Input data derives from published papers, studies and maps in FVG produced during the past 50 years and based on geological, geophysical, morphotectonic, seismological, paleoseismological data. The database is organized in direct or indirect clues on the location of active or presumably active faults, as well as on their recent activity (e.g. historical, Holocene, post-LGM, pre-LGM, Quaternary and Pre-Quaternary). These clues served to fix major tectonic structures, and to create the map of the active or presumably active faults. When data are available, the structures are characterized by a full set of geometric, kinematic and seismological parameters (e.g., slip rate, recurrence interval). Each parameter is rated for accuracy (e.g. quality index). We think that this database represents a useful tool to assess both ground shaking scenarios and surface faulting hazard.

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Fair Without Leveling Down: A New Intersectional Fairness Definition

Experimental Observations of the Topology of Convolutional Neural Network Activations

Hard magnet topological semimetals in XPt3 compounds with the harmony of Berry curvature

Box Embeddings: An open-source library for representation learning using geometric structures

Role of the long-lived Schio-Vicenza Fault System at the northern Adria plate margin

New evidence of recent tectonic activity of the Susans-Tricesimo thrust-system (NE Italy)

Harmonization of geological data in the Adriatic Sea

Landscape evolution and plate boundary processes: the study case of the Central Anatolian Plateau margins

Geodynamic and seismological modelling of the recent dynamics of the Central Mediterranean region

A multi-isotope (O, H, B, Sr) approach for identifying salinity contamination along the coastal sector of Murgia aquifer (Apulia, Southern Italy)

Complex deformation pattern of the October 2016 central Italy earthquakes from DInSAR data

Database of active faults in Friuli Venezia Giulia: a basic tool for seismic hazard assessment

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