Linear Pantographic Sheets: Existence and Uniqueness of Weak Solutions

The well-posedness of the boundary value problems for second gradient elasticity has been studied under the assumption of strong ellipticity of the dependence on the second placement gradients (see, e.g., Chambon and Moullet in Comput. Methods Appl. Mech. Eng. 193:2771–2796, 2004 and Mareno and Healey in SIAM J. Math. Anal. 38:103–115, 2006). The study of the equilibrium of planar pantographic lattices has been approached in two different ways: in dell’Isola et al. (Proc. R. Soc. Lond. Ser. A 472:20150, 2016) a discrete model was introduced involving extensional and rotational springs which is also valid in large deformations regimes while in Boutin et al. (Math. Mech. Complex Syst. 5:127–162, 2017) the lattice has been modelled as a set of beam elements interconnected by internal pivots, but the analysis was restricted to the linear case. In both papers a homogenized second gradient deformation energy, quadratic in the neighbourhood of non deformed configuration, is obtained via perturbative methods and the predictions obtained with the obtained continuum model are successfully compared with experiments. This energy is not strongly elliptic in its dependence on second gradients. We consider in this paper also the important particular case of pantographic lattices whose first gradient energy does not depend on shear deformation: this could be considered either a pathological case or an important exceptional case (see Stillwell et al. in Am. Math. Mon. 105:850–858, 1998 and Turro in Angew. Chem., Int. Ed. Engl. 39:2255–2259, 2000). In both cases we believe that such a particular case deserves some attention because of what we can understand by studying it (see Dyson in Science 200:677–678, 1978). This circumstance motivates the present paper, where we address the well-posedness of the planar linearized equilibrium problem for homogenized pantographic lattices. To do so: (i) we introduce a class of subsets of anisotropic Sobolev’s space as the most suitable energy space E relative to assigned boundary conditions; (ii) we prove that the considered strain energy density is coercive and positive definite in E; (iii) we prove that the set of placements for which the strain energy is vanishing (the so-called floppy modes) must strictly include rigid motions; (iv) we determine the restrictions on displacement boundary conditions which assure existence and uniqueness of linear static problems. The presented results represent one of the first mechanical applications of the concept of Anisotropic Sobolev space, initially introduced only on the basis of purely abstract mathematical considerations

Effect of top-down connections in Hierarchical Sparse Coding

Hierarchical Sparse Coding (HSC) is a powerful model to efficiently represent multi-dimensional, structured data such as images. The simplest solution to solve this computationally hard problem is to decompose it into independent layer-wise subproblems. However, neuroscientific evidence would suggest inter-connecting these subproblems as in the Predictive Coding (PC) theory, which adds top-down connections between consecutive layers. In this study, a new model called 2-Layers Sparse Predictive Coding (2L-SPC) is introduced to assess the impact of this inter-layer feedback connection. In particular, the 2L-SPC is compared with a Hierarchical Lasso (Hi-La) network made out of a sequence of independent Lasso layers. The 2L-SPC and the 2-layers Hi-La networks are trained on 4 different databases and with different sparsity parameters on each layer. First, we show that the overall prediction error generated by 2L-SPC is lower thanks to the feedback mechanism as it transfers prediction error between layers. Second, we demonstrate that the inference stage of the 2L-SPC is faster to converge than for the Hi-La model. Third, we show that the 2L-SPC also accelerates the learning process. Finally, the qualitative analysis of both models dictionaries, supported by their activation probability, show that the 2L-SPC features are more generic and informative

Diversity vs. Recognizability: Human-like generalization in one-shot generative models

Robust generalization to new concepts has long remained a distinctive feature of human intelligence. However, recent progress in deep generative models has now led to neural architectures capable of synthesizing novel instances of unknown visual concepts from a single training example. Yet, a more precise comparison between these models and humans is not possible because existing performance metrics for generative models (i.e., FID, IS, likelihood) are not appropriate for the one-shot generation scenario. Here, we propose a new framework to evaluate one-shot generative models along two axes: sample recognizability vs. diversity (i.e., intra-class variability). Using this framework, we perform a systematic evaluation of representative one-shot generative models on the Omniglot handwritten dataset. We first show that GAN-like and VAE-like models fall on opposite ends of the diversity-recognizability space. Extensive analyses of the effect of key model parameters further revealed that spatial attention and context integration have a linear contribution to the diversity-recognizability trade-off. In contrast, disentanglement transports the model along a parabolic curve that could be used to maximize recognizability. Using the diversity-recognizability framework, we were able to identify models and parameters that closely approximate human data

A Holistic Approach to Unifying Automatic Concept Extraction and Concept Importance Estimation

In recent years, concept-based approaches have emerged as some of the most promising explainability methods to help us interpret the decisions of Artificial Neural Networks (ANNs). These methods seek to discover intelligible visual 'concepts' buried within the complex patterns of ANN activations in two key steps: (1) concept extraction followed by (2) importance estimation. While these two steps are shared across methods, they all differ in their specific implementations. Here, we introduce a unifying theoretical framework that comprehensively defines and clarifies these two steps. This framework offers several advantages as it allows us: (i) to propose new evaluation metrics for comparing different concept extraction approaches; (ii) to leverage modern attribution methods and evaluation metrics to extend and systematically evaluate state-of-the-art concept-based approaches and importance estimation techniques; (iii) to derive theoretical guarantees regarding the optimality of such methods. We further leverage our framework to try to tackle a crucial question in explainability: how to efficiently identify clusters of data points that are classified based on a similar shared strategy. To illustrate these findings and to highlight the main strategies of a model, we introduce a visual representation called the strategic cluster graph. Finally, we present https://serre-lab.github.io/Lens, a dedicated website that offers a complete compilation of these visualizations for all classes of the ImageNet dataset

Unlocking Feature Visualization for Deeper Networks with MAgnitude Constrained Optimization

Feature visualization has gained substantial popularity, particularly after the influential work by Olah et al. in 2017, which established it as a crucial tool for explainability. However, its widespread adoption has been limited due to a reliance on tricks to generate interpretable images, and corresponding challenges in scaling it to deeper neural networks. Here, we describe MACO, a simple approach to address these shortcomings. The main idea is to generate images by optimizing the phase spectrum while keeping the magnitude constant to ensure that generated explanations lie in the space of natural images. Our approach yields significantly better results (both qualitatively and quantitatively) and unlocks efficient and interpretable feature visualizations for large state-of-the-art neural networks. We also show that our approach exhibits an attribution mechanism allowing us to augment feature visualizations with spatial importance. We validate our method on a novel benchmark for comparing feature visualization methods, and release its visualizations for all classes of the ImageNet dataset on https://serre-lab.github.io/Lens/. Overall, our approach unlocks, for the first time, feature visualizations for large, state-of-the-art deep neural networks without resorting to any parametric prior image model

Reconciling Apparent Conflicts between Mitochondrial and Nuclear Phylogenies in African Elephants

Conservation strategies for African elephants would be advanced by resolution of conflicting claims that they comprise one, two, three or four taxonomic groups, and by development of genetic markers that establish more incisively the provenance of confiscated ivory. We addressed these related issues by genotyping 555 elephants from across Africa with microsatellite markers, developing a method to identify those loci most effective at geographic assignment of elephants (or their ivory), and conducting novel analyses of continent-wide datasets of mitochondrial DNA. Results showed that nuclear genetic diversity was partitioned into two clusters, corresponding to African forest elephants (99.5% Cluster-1) and African savanna elephants (99.4% Cluster-2). Hybrid individuals were rare. In a comparison of basal forest “F” and savanna “S” mtDNA clade distributions to nuclear DNA partitions, forest elephant nuclear genotypes occurred only in populations in which S clade mtDNA was absent, suggesting that nuclear partitioning corresponds to the presence or absence of S clade mtDNA. We reanalyzed African elephant mtDNA sequences from 81 locales spanning the continent and discovered that S clade mtDNA was completely absent among elephants at all 30 sampled tropical forest locales. The distribution of savanna nuclear DNA and S clade mtDNA corresponded closely to range boundaries traditionally ascribed to the savanna elephant species based on habitat and morphology. Further, a reanalysis of nuclear genetic assignment results suggested that West African elephants do not comprise a distinct third species. Finally, we show that some DNA markers will be more useful than others for determining the geographic origins of illegal ivory. These findings resolve the apparent incongruence between mtDNA and nuclear genetic patterns that has confounded the taxonomy of African elephants, affirm the limitations of using mtDNA patterns to infer elephant systematics or population structure, and strongly support the existence of two elephant species in Africa