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Nanopore sequencing is a promising technology for DNA sequencing. In this paper, we investigate a specific model of the nanopore sequencer, which takes a $q$-ary sequence of length $n$ as input and outputs a vector of length $n+\ell-1$ referred to as an $\ell$-read vector where the $i$-th entry is a multi-set composed of the $\ell$ elements located between the $(i-\ell+1)$-th and $i$-th positions of the input sequence. Considering the presence of substitution errors in the output vector, we study $\ell$-read codes under the Hamming metric. An $\ell$-read $(n,d)_q$-code is a set of $q$-ary sequences of length $n$ in which the Hamming distance between $\ell$-read vectors of any two distinct sequences is at least $d$. We first improve the result of Banerjee \emph{et al.}, who studied $\ell$-read $(n,d)_q$-codes with the constraint $\ell\geq 3$ and $d=3$. Then, we investigate the bounds and constructions of $2$-read codes with a minimum distance of $3$, $4$, and $5$, respectively. Our results indicate that when $d \in \{3,4\}$, the optimal redundancy of $2$-read $(n,d)_q$-codes is $o(\log_q n)$, while for $d=5$ it is $\log_q n+o(\log_q n)$. Additionally, we establish an equivalence between $2$-read $(n,3)_q$-codes and classical $q$-ary single-insertion reconstruction codes using two noisy reads. We improve the lower bound on the redundancy of classical $q$-ary single-insertion reconstruction codes as well as the upper bound on the redundancy of classical $q$-ary single-deletion reconstruction codes when using two noisy reads. Finally, we study $\ell$-read codes under the reconstruction model.

In this paper, we approach the problem of cost (loss) minimization in underparametrized shallow neural networks through the explicit construction of upper bounds, without any use of gradient descent. A key focus is on elucidating the geometric structure of approximate and precise minimizers. We consider shallow neural networks with one hidden layer, a ReLU activation function, an ${\mathcal L}^2$ Schatten class (or Hilbert-Schmidt) cost function, input space ${\mathbb R}^M$, output space ${\mathbb R}^Q$ with $Q\leq M$, and training input sample size $N>QM$ that can be arbitrarily large. We prove an upper bound on the minimum of the cost function of order $O(\delta_P)$ where $\delta_P$ measures the signal to noise ratio of training inputs. In the special case $M=Q$, we explicitly determine an exact degenerate local minimum of the cost function, and show that the sharp value differs from the upper bound obtained for $Q\leq M$ by a relative error $O(\delta_P^2)$. The proof of the upper bound yields a constructively trained network; we show that it metrizes a particular $Q$-dimensional subspace in the input space ${\mathbb R}^M$. We comment on the characterization of the global minimum of the cost function in the given context.

In this paper, we explicitly determine local and global minimizers of the $\mathcal{L}^2$ cost function in underparametrized Deep Learning (DL) networks; our main goal is to shed light on their geometric structure and properties. We accomplish this by a direct construction, without invoking the gradient descent flow at any point of this work. We specifically consider $L$ hidden layers, a ReLU ramp activation function, an $\mathcal{L}^2$ Schatten class (or Hilbert-Schmidt) cost function, input and output spaces $\mathbb{R}^Q$ with equal dimension $Q\geq1$, and hidden layers also defined on $\mathbb{R}^{Q}$; the training inputs are assumed to be sufficiently clustered. The training input size $N$ can be arbitrarily large - thus, we are considering the underparametrized regime. More general settings are left to future work. We construct an explicit family of minimizers for the global minimum of the cost function in the case $L\geq Q$, which we show to be degenerate. Moreover, we determine a set of $2^Q-1$ distinct degenerate local minima of the cost function. In the context presented here, the concatenation of hidden layers of the DL network is reinterpreted as a recursive application of a {\em truncation map} which "curates" the training inputs by minimizing their noise to signal ratio.

The increasing reliance on numerical methods for controlling dynamical systems and training machine learning models underscores the need to devise algorithms that dependably and efficiently navigate complex optimization landscapes. Classical gradient descent methods offer strong theoretical guarantees for convex problems; however, they demand meticulous hyperparameter tuning for non-convex ones. The emerging paradigm of learning to optimize (L2O) automates the discovery of algorithms with optimized performance leveraging learning models and data - yet, it lacks a theoretical framework to analyze convergence and robustness of the learned algorithms. In this paper, we fill this gap by harnessing nonlinear system theory. Specifically, we propose an unconstrained parametrization of all convergent algorithms for smooth non-convex objective functions. Notably, our framework is directly compatible with automatic differentiation tools, ensuring convergence by design while learning to optimize.

This work presents a comparative review and classification between some well-known thermodynamically consistent models of hydrogel behavior in a large deformation setting, specifically focusing on solvent absorption/desorption and its impact on mechanical deformation and network swelling. The proposed discussion addresses formulation aspects, general mathematical classification of the governing equations, and numerical implementation issues based on the finite element method. The theories are presented in a unified framework demonstrating that, despite not being evident in some cases, all of them follow equivalent thermodynamic arguments. A detailed numerical analysis is carried out where Taylor-Hood elements are employed in the spatial discretization to satisfy the inf-sup condition and to prevent spurious numerical oscillations. The resulting discrete problems are solved using the FEniCS platform through consistent variational formulations, employing both monolithic and staggered approaches. We conduct benchmark tests on various hydrogel structures, demonstrating that major differences arise from the chosen volumetric response of the hydrogel. The significance of this choice is frequently underestimated in the state-of-the-art literature but has been shown to have substantial implications on the resulting hydrogel behavior.

Explicit time integration schemes coupled with Galerkin discretizations of time-dependent partial differential equations require solving a linear system with the mass matrix at each time step. For applications in structural dynamics, the solution of the linear system is frequently approximated through so-called mass lumping, which consists in replacing the mass matrix by some diagonal approximation. Mass lumping has been widely used in engineering practice for decades already and has a sound mathematical theory supporting it for finite element methods using the classical Lagrange basis. However, the theory for more general basis functions is still missing. Our paper partly addresses this shortcoming. Some special and practically relevant properties of lumped mass matrices are proved and we discuss how these properties naturally extend to banded and Kronecker product matrices whose structure allows to solve linear systems very efficiently. Our theoretical results are applied to isogeometric discretizations but are not restricted to them.

Object detection on visible (RGB) and infrared (IR) images, as an emerging solution to facilitate robust detection for around-the-clock applications, has received extensive attention in recent years. With the help of IR images, object detectors have been more reliable and robust in practical applications by using RGB-IR combined information. However, existing methods still suffer from modality miscalibration and fusion imprecision problems. Since transformer has the powerful capability to model the pairwise correlations between different features, in this paper, we propose a novel Calibrated and Complementary Transformer called $\mathrm{C}^2$Former to address these two problems simultaneously. In $\mathrm{C}^2$Former, we design an Inter-modality Cross-Attention (ICA) module to obtain the calibrated and complementary features by learning the cross-attention relationship between the RGB and IR modality. To reduce the computational cost caused by computing the global attention in ICA, an Adaptive Feature Sampling (AFS) module is introduced to decrease the dimension of feature maps. Because $\mathrm{C}^2$Former performs in the feature domain, it can be embedded into existed RGB-IR object detectors via the backbone network. Thus, one single-stage and one two-stage object detector both incorporating our $\mathrm{C}^2$Former are constructed to evaluate its effectiveness and versatility. With extensive experiments on the DroneVehicle and KAIST RGB-IR datasets, we verify that our method can fully utilize the RGB-IR complementary information and achieve robust detection results. The code is available at //github.com/yuanmaoxun/Calibrated-and-Complementary-Transformer-for-RGB-Infrared-Object-Detection.git.

Constructing small-sized coresets for various clustering problems in different metric spaces has attracted significant attention for the past decade. A central problem in the coreset literature is to understand what is the best possible coreset size for $(k,z)$-clustering in Euclidean space. While there has been significant progress in the problem, there is still a gap between the state-of-the-art upper and lower bounds. For instance, the best known upper bound for $k$-means ($z=2$) is $\min \{O(k^{3/2} \varepsilon^{-2}),O(k \varepsilon^{-4})\}$ [1,2], while the best known lower bound is $\Omega(k\varepsilon^{-2})$ [1]. In this paper, we make significant progress on both upper and lower bounds. For a large range of parameters (i.e., $\varepsilon, k$), we have a complete understanding of the optimal coreset size. In particular, we obtain the following results: (1) We present a new coreset lower bound $\Omega(k \varepsilon^{-z-2})$ for Euclidean $(k,z)$-clustering when $\varepsilon \geq \Omega(k^{-1/(z+2)})$. In view of the prior upper bound $\tilde{O}_z(k \varepsilon^{-z-2})$ [1], the bound is optimal. The new lower bound also implies improved lower bounds for $(k,z)$-clustering in doubling metrics. (2) For the upper bound, we provide efficient coreset construction algorithms for $(k,z)$-clustering with improved or optimal coreset sizes in several metric spaces. In particular, we provide an $\tilde{O}_z(k^{\frac{2z+2}{z+2}} \varepsilon^{-2})$-sized coreset, with a unfied analysis, for $(k,z)$-clustering for all $z\geq 1$ in Euclidean space. [1] Cohen-Addad, Larsen, Saulpic, Schwiegelshohn. STOC'22. [2] Cohen-Addad, Larsen, Saulpic, Schwiegelshohn, Sheikh-Omar, NeurIPS'22.

This paper introduces AL$\ell_0$CORE, a new form of probabilistic non-negative tensor decomposition. AL$\ell_0$CORE is a Tucker decomposition where the number of non-zero elements (i.e., the $\ell_0$-norm) of the core tensor is constrained to a preset value $Q$ much smaller than the size of the core. While the user dictates the total budget $Q$, the locations and values of the non-zero elements are latent variables and allocated across the core tensor during inference. AL$\ell_0$CORE -- i.e., $allo$cated $\ell_0$-$co$nstrained $core$-- thus enjoys both the computational tractability of CP decomposition and the qualitatively appealing latent structure of Tucker. In a suite of real-data experiments, we demonstrate that AL$\ell_0$CORE typically requires only tiny fractions (e.g.,~1%) of the full core to achieve the same results as full Tucker decomposition at only a correspondingly tiny fraction of the cost.

This paper proposes a~simple, yet powerful, method for balancing distributions of covariates for causal inference based on observational studies. The method makes it possible to balance an arbitrary number of quantiles (e.g., medians, quartiles, or deciles) together with means if necessary. The proposed approach is based on the theory of calibration estimators (Deville and S\"arndal 1992), in particular, calibration estimators for quantiles, proposed by Harms and Duchesne (2006). The method does not require numerical integration, kernel density estimation or assumptions about the distributions. Valid estimates can be obtained by drawing on existing asymptotic theory. An~illustrative example of the proposed approach is presented for the entropy balancing method and the covariate balancing propensity score method. Results of a~simulation study indicate that the method efficiently estimates average treatment effects on the treated (ATT), the average treatment effect (ATE), the quantile treatment effect on the treated (QTT) and the quantile treatment effect (QTE), especially in the presence of non-linearity and mis-specification of the models. The proposed approach can be further generalized to other designs (e.g. multi-category, continuous) or methods (e.g. synthetic control method). An open source software implementing proposed methods is available.