We prove a far-reaching strengthening of Szemer\'edi's regularity lemma for intersection graphs of pseudo-segments. It shows that the vertex set of such a graph can be partitioned into a bounded number of parts of roughly the same size such that almost all bipartite graphs between different pairs of parts are complete or empty. We use this to get an improved bound on disjoint edges in simple topological graphs, showing that every $n$-vertex simple topological graph with no $k$ pairwise disjoint edges has at most $n(\log n)^{O(\log k)}$ edges.
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Epilepsy is a clinical neurological disorder characterized by recurrent and spontaneous seizures consisting of abnormal high-frequency electrical activity in the brain. In this condition, the transmembrane potential dynamics are characterized by rapid and sharp wavefronts traveling along the heterogeneous and anisotropic conduction pathways of the brain. This work employs the monodomain model, coupled with specific neuronal ionic models characterizing ion concentration dynamics, to mathematically describe brain tissue electrophysiology in grey and white matter at the organ scale. This multiscale model is discretized in space with the high-order discontinuous Galerkin method on polygonal and polyhedral grids (PolyDG) and advanced in time with a Crank-Nicolson scheme. This ensures, on the one hand, efficient and accurate simulations of the high-frequency electrical activity that is responsible for epileptic seizure and, on the other hand, keeps reasonably low the computational costs by a suitable combination of high-order approximations and agglomerated polytopal meshes. We numerically investigate synthetic test cases on a two-dimensional heterogeneous squared domain discretized with a polygonal grid, and on a two-dimensional brainstem in a sagittal plane with an agglomerated polygonal grid that takes full advantage of the flexibility of the PolyDG approximation of the semidiscrete formulation. Finally, we provide a theoretical analysis of stability and an a-priori convergence analysis for a simplified mathematical problem.
We propose a two-step Newton's method for refining an approximation of a singular zero whose deflation process terminates after one step, also known as a deflation-one singularity. Given an isolated singular zero of a square analytic system, our algorithm exploits an invertible linear operator obtained by combining the Jacobian and a projection of the Hessian in the direction of the kernel of the Jacobian. We prove the quadratic convergence of the two-step Newton method when it is applied to an approximation of a deflation-one singular zero. Also, the algorithm requires a smaller size of matrices than the existing methods, making it more efficient. We demonstrate examples and experiments to show the efficiency of the method.
The utility of a learned neural representation depends on how well its geometry supports performance in downstream tasks. This geometry depends on the structure of the inputs, the structure of the target outputs, and the architecture of the network. By studying the learning dynamics of networks with one hidden layer, we discovered that the network's activation function has an unexpectedly strong impact on the representational geometry: Tanh networks tend to learn representations that reflect the structure of the target outputs, while ReLU networks retain more information about the structure of the raw inputs. This difference is consistently observed across a broad class of parameterized tasks in which we modulated the degree of alignment between the geometry of the task inputs and that of the task labels. We analyzed the learning dynamics in weight space and show how the differences between the networks with Tanh and ReLU nonlinearities arise from the asymmetric asymptotic behavior of ReLU, which leads feature neurons to specialize for different regions of input space. By contrast, feature neurons in Tanh networks tend to inherit the task label structure. Consequently, when the target outputs are low dimensional, Tanh networks generate neural representations that are more disentangled than those obtained with a ReLU nonlinearity. Our findings shed light on the interplay between input-output geometry, nonlinearity, and learned representations in neural networks.
Generating series are crucial in enumerative combinatorics, analytic combinatorics, and combinatorics on words. Though it might seem at first view that generating Dirichlet series are less used in these fields than ordinary and exponential generating series, there are many notable papers where they play a fundamental role, as can be seen in particular in the work of Flajolet and several of his co-authors. In this paper, we study Dirichlet series of integers with missing digits or blocks of digits in some integer base $b$, i.e., where the summation ranges over the integers whose expansions form some language strictly included in the set of all words on the alphabet $\{0, 1, \dots, b-1\}$ that do not begin with a $0$. We show how to unify and extend results proved by Nathanson in 2021 and by K\"ohler and Spilker in 2009. En route, we encounter several sequences from Sloane's On-Line Encyclopedia of Integer Sequences, as well as some famous $q$-automatic sequences or $q$-regular sequences.
A generalization of Passing-Bablok regression is proposed for comparing multiple measurement methods simultaneously. Possible applications include assay migration studies or interlaboratory trials. When comparing only two methods, the method reduces to the usual Passing-Bablok estimator. It is close in spirit to reduced major axis regression, which is, however, not robust. To obtain a robust estimator, the major axis is replaced by the (hyper-)spherical median axis. The method is shown to reduce to the usual Passing-Bablok estimator if only two methods are compared. This technique has been applied to compare SARS-CoV-2 serological tests, bilirubin in neonates, and an in vitro diagnostic test using different instruments, sample preparations, and reagent lots. In addition, plots similar to the well-known Bland-Altman plots have been developed to represent the variance structure.
Right-linear (or left-linear) grammars are a well-known class of context-free grammars computing just the regular languages. They may naturally be written as expressions with (least) fixed points but with products restricted to letters as left arguments, giving an alternative to the syntax of regular expressions. In this work, we investigate the resulting logical theory of this syntax. Namely, we propose a theory of right-linear algebras (RLA) over of this syntax and a cyclic proof system CRLA for reasoning about them. We show that CRLA is sound and complete for the intended model of regular languages. From here we recover the same completeness result for RLA by extracting inductive invariants from cyclic proofs, rendering the model of regular languages the free right-linear algebra. Finally, we extend system CRLA by greatest fixed points, nuCRLA, naturally modelled by languages of omega-words thanks to right-linearity. We show a similar soundness and completeness result of (the guarded fragment of) nuCRLA for the model of omega-regular languages, employing game theoretic techniques.
Remotely sensed data are dominated by mixed Land Use and Land Cover (LULC) types. Spectral unmixing (SU) is a key technique that disentangles mixed pixels into constituent LULC types and their abundance fractions. While existing studies on Deep Learning (DL) for SU typically focus on single time-step hyperspectral (HS) or multispectral (MS) data, our work pioneers SU using MODIS MS time series, addressing missing data with end-to-end DL models. Our approach enhances a Long-Short Term Memory (LSTM)-based model by incorporating geographic, topographic (geo-topographic), and climatic ancillary information. Notably, our method eliminates the need for explicit endmember extraction, instead learning the input-output relationship between mixed spectra and LULC abundances through supervised learning. Experimental results demonstrate that integrating spectral-temporal input data with geo-topographic and climatic information significantly improves the estimation of LULC abundances in mixed pixels. To facilitate this study, we curated a novel labeled dataset for Andalusia (Spain) with monthly MODIS multispectral time series at 460m resolution for 2013. Named Andalusia MultiSpectral MultiTemporal Unmixing (Andalusia-MSMTU), this dataset provides pixel-level annotations of LULC abundances along with ancillary information. The dataset (//zenodo.org/records/7752348) and code (//github.com/jrodriguezortega/MSMTU) are available to the public.
It is well known that for singular inconsistent range-symmetric linear systems, the generalized minimal residual (GMRES) method determines a least squares solution without breakdown. The reached least squares solution may be or not be the pseudoinverse solution. We show that a lift strategy can be used to obtain the pseudoinverse solution. In addition, we propose a new iterative method named RSMAR (minimum $\mathbf A$-residual) for range-symmetric linear systems $\mathbf A\mathbf x=\mathbf b$. At step $k$ RSMAR minimizes $\|\mathbf A\mathbf r_k\|$ in the $k$th Krylov subspace generated with $\{\mathbf A, \mathbf r_0\}$ rather than $\|\mathbf r_k\|$, where $\mathbf r_k$ is the $k$th residual vector and $\|\cdot\|$ denotes the Euclidean vector norm. We show that RSMAR and GMRES terminate with the same least squares solution when applied to range-symmetric linear systems. We provide two implementations for RSMAR. Our numerical experiments show that RSMAR is the most suitable method among GMRES-type methods for singular inconsistent range-symmetric linear systems.
The goal of explainable Artificial Intelligence (XAI) is to generate human-interpretable explanations, but there are no computationally precise theories of how humans interpret AI generated explanations. The lack of theory means that validation of XAI must be done empirically, on a case-by-case basis, which prevents systematic theory-building in XAI. We propose a psychological theory of how humans draw conclusions from saliency maps, the most common form of XAI explanation, which for the first time allows for precise prediction of explainee inference conditioned on explanation. Our theory posits that absent explanation humans expect the AI to make similar decisions to themselves, and that they interpret an explanation by comparison to the explanations they themselves would give. Comparison is formalized via Shepard's universal law of generalization in a similarity space, a classic theory from cognitive science. A pre-registered user study on AI image classifications with saliency map explanations demonstrate that our theory quantitatively matches participants' predictions of the AI.
We hypothesize that due to the greedy nature of learning in multi-modal deep neural networks, these models tend to rely on just one modality while under-fitting the other modalities. Such behavior is counter-intuitive and hurts the models' generalization, as we observe empirically. To estimate the model's dependence on each modality, we compute the gain on the accuracy when the model has access to it in addition to another modality. We refer to this gain as the conditional utilization rate. In the experiments, we consistently observe an imbalance in conditional utilization rates between modalities, across multiple tasks and architectures. Since conditional utilization rate cannot be computed efficiently during training, we introduce a proxy for it based on the pace at which the model learns from each modality, which we refer to as the conditional learning speed. We propose an algorithm to balance the conditional learning speeds between modalities during training and demonstrate that it indeed addresses the issue of greedy learning. The proposed algorithm improves the model's generalization on three datasets: Colored MNIST, Princeton ModelNet40, and NVIDIA Dynamic Hand Gesture.
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