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As machine learning becomes increasingly prevalent in impactful decisions, recognizing when inference data is outside the model's expected input distribution is paramount for giving context to predictions. Out-of-distribution (OOD) detection methods have been created for this task. Such methods can be split into representation-based or logit-based methods from whether they respectively utilize the model's embeddings or predictions for OOD detection. In contrast to most papers which solely focus on one such group, we address both. We employ dimensionality reduction on feature embeddings in representation-based methods for both time speedups and improved performance. Additionally, we propose DICE-COL, a modification of the popular logit-based method Directed Sparsification (DICE) that resolves an unnoticed flaw. We demonstrate the effectiveness of our methods on the OpenOODv1.5 benchmark framework, where they significantly improve performance and set state-of-the-art results.

Data consisting of a graph with a function to $\mathbb{R}^d$ arise in many data applications, encompassing structures such as Reeb graphs, geometric graphs, and knot embeddings. As such, the ability to compare and cluster such objects is required in a data analysis pipeline, leading to a need for distances or metrics between them. In this work, we study the interleaving distance on discretizations of these objects, $\mathbb{R}^d$-mapper graphs, where functor representations of the data can be compared by finding pairs of natural transformations between them. However, in many cases, computation of the interleaving distance is NP-hard. For this reason, we take inspiration from the work of Robinson to find quality measures for families of maps that do not rise to the level of a natural transformation, called assignments. We then endow the functor images with the extra structure of a metric space and define a loss function which measures how far an assignment is from making the required diagrams of an interleaving commute. Finally we show that the computation of the loss function is polynomial. We believe this idea is both powerful and translatable, with the potential to be used for approximation and bounds on interleavings in a broad array of contexts.

Graph learning architectures based on the k-dimensional Weisfeiler-Leman (k-WL) hierarchy offer a theoretically well-understood expressive power. However, such architectures often fail to deliver solid predictive performance on real-world tasks, limiting their practical impact. In contrast, global attention-based models such as graph transformers demonstrate strong performance in practice, but comparing their expressive power with the k-WL hierarchy remains challenging, particularly since these architectures rely on positional or structural encodings for their expressivity and predictive performance. To address this, we show that the recently proposed Edge Transformer, a global attention model operating on node pairs instead of nodes, has at least 3-WL expressive power. Empirically, we demonstrate that the Edge Transformer surpasses other theoretically aligned architectures regarding predictive performance while not relying on positional or structural encodings.

Interpolatory necessary optimality conditions for $\mathcal{H}_2$-optimal reduced-order modeling of non-parametric linear time-invariant (LTI) systems are known and well-investigated. In this work, using the general framework of $\mathcal{L}_2$-optimal reduced-order modeling of parametric stationary problems, we derive interpolatory $\mathcal{H}_2 \otimes \mathcal{L}_2$-optimality conditions for parametric LTI systems with a general pole-residue form. We then specialize this result to recover known conditions for systems with parameter-independent poles and develop new conditions for a certain class of systems with parameter-dependent poles.

Loss function learning is a new meta-learning paradigm that aims to automate the essential task of designing a loss function for a machine learning model. Existing techniques for loss function learning have shown promising results, often improving a model's training dynamics and final inference performance. However, a significant limitation of these techniques is that the loss functions are meta-learned in an offline fashion, where the meta-objective only considers the very first few steps of training, which is a significantly shorter time horizon than the one typically used for training deep neural networks. This causes significant bias towards loss functions that perform well at the very start of training but perform poorly at the end of training. To address this issue we propose a new loss function learning technique for adaptively updating the loss function online after each update to the base model parameters. The experimental results show that our proposed method consistently outperforms the cross-entropy loss and offline loss function learning techniques on a diverse range of neural network architectures and datasets.

Traditionally, classical numerical schemes have been employed to solve partial differential equations (PDEs) using computational methods. Recently, neural network-based methods have emerged. Despite these advancements, neural network-based methods, such as physics-informed neural networks (PINNs) and neural operators, exhibit deficiencies in robustness and generalization. To address these issues, numerous studies have integrated classical numerical frameworks with machine learning techniques, incorporating neural networks into parts of traditional numerical methods. In this study, we focus on hyperbolic conservation laws by replacing traditional numerical fluxes with neural operators. To this end, we developed loss functions inspired by established numerical schemes related to conservation laws and approximated numerical fluxes using Fourier neural operators (FNOs). Our experiments demonstrated that our approach combines the strengths of both traditional numerical schemes and FNOs, outperforming standard FNO methods in several respects. For instance, we demonstrate that our method is robust, has resolution invariance, and is feasible as a data-driven method. In particular, our method can make continuous predictions over time and exhibits superior generalization capabilities with out-of-distribution (OOD) samples, which are challenges that existing neural operator methods encounter.

We propose a new framework to design and analyze accelerated methods that solve general monotone equation (ME) problems $F(x)=0$. Traditional approaches include generalized steepest descent methods and inexact Newton-type methods. If $F$ is uniformly monotone and twice differentiable, these methods achieve local convergence rates while the latter methods are globally convergent thanks to line search and hyperplane projection. However, a global rate is unknown for these methods. The variational inequality methods can be applied to yield a global rate that is expressed in terms of $\|F(x)\|$ but these results are restricted to first-order methods and a Lipschitz continuous operator. It has not been clear how to obtain global acceleration using high-order Lipschitz continuity. This paper takes a continuous-time perspective where accelerated methods are viewed as the discretization of dynamical systems. Our contribution is to propose accelerated rescaled gradient systems and prove that they are equivalent to closed-loop control systems. Based on this connection, we establish the properties of solution trajectories. Moreover, we provide a unified algorithmic framework obtained from discretization of our system, which together with two approximation subroutines yields both existing high-order methods and new first-order methods. We prove that the $p^{th}$-order method achieves a global rate of $O(k^{-p/2})$ in terms of $\|F(x)\|$ if $F$ is $p^{th}$-order Lipschitz continuous and the first-order method achieves the same rate if $F$ is $p^{th}$-order strongly Lipschitz continuous. If $F$ is strongly monotone, the restarted versions achieve local convergence with order $p$ when $p \geq 2$. Our discrete-time analysis is largely motivated by the continuous-time analysis and demonstrates the fundamental role that rescaled gradients play in global acceleration for solving ME problems.

We extend Petkov\v{s}ek's algorithm for computing hypergeometric solutions of scalar difference equations to the case of difference systems $\tau(Y) = M Y$, with $M \in {\rm GL}_n(C(x))$, where $\tau$ is the shift operator. Hypergeometric solutions are solutions of the form $\gamma P$ where $P \in C(x)^n$ and $\gamma$ is a hypergeometric term over $C(x)$, i.e. ${\tau(\gamma)}/{\gamma} \in C(x)$. Our contributions concern efficient computation of a set of candidates for ${\tau(\gamma)}/{\gamma}$ which we write as $\lambda = c\frac{A}{B}$ with monic $A, B \in C[x]$, $c \in C^*$. Factors of the denominators of $M^{-1}$ and $M$ give candidates for $A$ and $B$, while another algorithm is needed for $c$. We use the super-reduction algorithm to compute candidates for $c$, as well as other ingredients to reduce the list of candidates for $A/B$. To further reduce the number of candidates $A/B$, we bound the so-called type of $A/B$ by bounding local types. Our algorithm has been implemented in Maple and experiments show that our implementation can handle systems of high dimension, which is useful for factoring operators.

Many machine learning approaches for decision making, such as reinforcement learning, rely on simulators or predictive models to forecast the time-evolution of quantities of interest, e.g., the state of an agent or the reward of a policy. Forecasts of such complex phenomena are commonly described by highly nonlinear dynamical systems, making their use in optimization-based decision-making challenging. Koopman operator theory offers a beneficial paradigm for addressing this problem by characterizing forecasts via linear time-invariant (LTI) ODEs, turning multi-step forecasts into sparse matrix multiplication. Though there exists a variety of learning approaches, they usually lack crucial learning-theoretic guarantees, making the behavior of the obtained models with increasing data and dimensionality unclear. We address the aforementioned by deriving a universal Koopman-invariant reproducing kernel Hilbert space (RKHS) that solely spans transformations into LTI dynamical systems. The resulting Koopman Kernel Regression (KKR) framework enables the use of statistical learning tools from function approximation for novel convergence results and generalization error bounds under weaker assumptions than existing work. Our experiments demonstrate superior forecasting performance compared to Koopman operator and sequential data predictors in RKHS.

We investigate a lattice-structured LSTM model for Chinese NER, which encodes a sequence of input characters as well as all potential words that match a lexicon. Compared with character-based methods, our model explicitly leverages word and word sequence information. Compared with word-based methods, lattice LSTM does not suffer from segmentation errors. Gated recurrent cells allow our model to choose the most relevant characters and words from a sentence for better NER results. Experiments on various datasets show that lattice LSTM outperforms both word-based and character-based LSTM baselines, achieving the best results.