This paper presents a task-oriented computational framework to enhance Visual-Inertial Navigation (VIN) in robots, addressing challenges such as limited time and energy resources. The framework strategically selects visual features using a Mean Squared Error (MSE)-based, non-submodular objective function and a simplified dynamic anticipation model. To address the NP-hardness of this problem, we introduce four polynomial-time approximation algorithms: a classic greedy method with constant-factor guarantees; a low-rank greedy variant that significantly reduces computational complexity; a randomized greedy sampler that balances efficiency and solution quality; and a linearization-based selector based on a first-order Taylor expansion for near-constant-time execution. We establish rigorous performance bounds by leveraging submodularity ratios, curvature, and element-wise curvature analyses. Extensive experiments on both standardized benchmarks and a custom control-aware platform validate our theoretical results, demonstrating that these methods achieve strong approximation guarantees while enabling real-time deployment.

In-context learning (ICL) has emerged as a powerful paradigm for adapting large language models (LLMs) to new and data-scarce tasks using only a few carefully selected task-specific examples presented in the prompt. However, given the limited context size of LLMs, a fundamental question arises: Which examples should be selected to maximize performance on a given user query? While nearest-neighbor-based methods like KATE have been widely adopted for this purpose, they suffer from well-known drawbacks in high-dimensional embedding spaces, including poor generalization and a lack of diversity. In this work, we study this problem of example selection in ICL from a principled, information theory-driven perspective. We first model an LLM as a linear function over input embeddings and frame the example selection task as a query-specific optimization problem: selecting a subset of exemplars from a larger example bank that minimizes the prediction error on a specific query. This formulation departs from traditional generalization-focused learning theoretic approaches by targeting accurate prediction for a specific query instance. We derive a principled surrogate objective that is approximately submodular, enabling the use of a greedy algorithm with an approximation guarantee. We further enhance our method by (i) incorporating the kernel trick to operate in high-dimensional feature spaces without explicit mappings, and (ii) introducing an optimal design-based regularizer to encourage diversity in the selected examples. Empirically, we demonstrate significant improvements over standard retrieval methods across a suite of classification tasks, highlighting the benefits of structure-aware, diverse example selection for ICL in real-world, label-scarce scenarios.

There are really two issues with this paper. The first one is the discussion on how to use directed information for causality. Unfortunately, the word causality means two different things and these should be separated (and this discussion should be mentioned in this manuscript I think). The first concept of causality is really *prediction* of a time series using other time series. This prediction respects time and hence people call that causal.

This paper considers the problem of maximizing a monotone set function subject to a cardinality constraint. The authors consider a novel combination of functions with both bounded submodularity ratio and additive noise. These setting have been considered separately before, but a joint analysis leads to a novel algorithm PONSS. This has improved theoretical guarantees and experimental performance when compared to previous noise-agnostic greedy algorithms. The paper flows well and is generally a pleasure to read.

We consider the problem of selecting an optimal subset of information sources for a hypothesis testing/classification task where the goal is to identify the true state of the world from a finite set of hypotheses, based on finite observation samples from the sources. In order to characterize the learning performance, we propose a misclassification penalty framework, which enables nonuniform treatment of different misclassification errors. In a centralized Bayesian learning setting, we study two variants of the subset selection problem: (i) selecting a minimum cost information set to ensure that the maximum penalty of misclassifying the true hypothesis is below a desired bound and (ii) selecting an optimal information set under a limited budget to minimize the maximum penalty of misclassifying the true hypothesis. Under certain assumptions, we prove that the objective (or constraints) of these combinatorial optimization problems are weak (or approximate) submodular, and establish high-probability performance guarantees for greedy algorithms. Further, we propose an alternate metric for information set selection which is based on the total penalty of misclassification. We prove that this metric is submodular and establish near-optimal guarantees for the greedy algorithms for both the information set selection problems. Finally, we present numerical simulations to validate our theoretical results over several randomly generated instances.

We study the problem of diverse feature selection in linear regression: selecting a small subset of diverse features that can predict a given objective. Diversity is useful for several reasons such as interpretability, robustness to noise, etc. We propose several spectral regularizers that capture a notion of diversity of features and show that these are all submodular set functions. These regularizers, when added to the objective function for linear regression, result in approximately submodular functions, which can then be maximized by efficient greedy and local search algorithms, with provable guarantees.