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   Acosta, Francisco; Conwell, Colin; Sanborn, Sophia; Klindt, David; Miolane, Nina (December 2023, NeurIPS Workshop on Unifying Representations)

   A fundamental principle of neural representation is to minimize wiring length by spatially organizing neurons according to the frequency of their communication [Sterling and Laughlin, 2015]. A consequence is that nearby regions of the brain tend to represent similar content. This has been explored in the context of the visual cortex in recent works [Doshi and Konkle, 2023, Tong et al., 2023]. Here, we use the notion of cortical distance as a baseline to ground, evaluate, and interpret measures of representational distance. We compare several popular methods—both second-order methods (Representational Similarity Analysis, Centered Kernel Alignment) and first-order methods (Shape Metrics)—and calculate how well the representational distance reflects 2D anatomical distance along the visual cortex (the anatomical stress score). We evaluate these metrics on a large-scale fMRI dataset of human ventral visual cortex [Allen et al., 2022b], and observe that the 3 types of Shape Metrics produce representational-anatomical stress scores with the smallest variance across subjects, (Z score = -1.5), which suggests that first-order representational scores quantify the relationship between representational and cortical geometry in a way that is more invariant across different subjects. Our work establishes a criterion with which to compare methods for quantifying representational similarity with implications for studying the anatomical organization of high-level ventral visual cortex. 

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3. Stratified Noncommutative Geometry

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   We introduce a theory of stratifications of noncommutative stacks (i.e., presentable stable $\mathrm{\infty <\#comment/>}$-categories), and we prove a reconstruction theorem that expresses them in terms of their strata and gluing data. This reconstruction theorem is compatible with symmetric monoidal structures, and with more general operadic structures such as ${\mathbb{E}}_n$-monoidal structures. We also provide a suite of fundamental operations for constructing new stratifications from old ones: restriction, pullback, quotient, pushforward, and refinement. Moreover, we establish a dual form of reconstruction; this is closely related to Verdier duality and reflection functors, and gives a categorification of Möbius inversion. Our main application is to equivariant stable homotopy theory: for any compact Lie group $G$, we give a symmetric monoidal stratification of genuine $G$-spectra. In the case that $G$is finite, this expresses genuine $G$-spectra in terms of their geometric fixedpoints (as homotopy-equivariant spectra) and gluing data therebetween (which are given by proper Tate constructions). We also prove an adelic reconstruction theorem; this applies not just to ordinary schemes but in the more general context of tensor-triangular geometry, where we obtain a symmetric monoidal stratification over the Balmer spectrum. We discuss the particular example of chromatic homotopy theory. 

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4. Geometry of moduli

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5. Geometry Regularized Autoencoders

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   Duque, Andres F.; Morin, Sacha; Wolf, Guy; Moon, Kevin R. (June 2023, IEEE Transactions on Pattern Analysis and Machine Intelligence)