We develop a novel approach for Linear combination of unitaries (LCU) type decomposition for structured sparse matrices. Such matrices frequently arise during numerical solution of partial differential equations which are ubiquitous in science and engineering. LCU is a versatile quantum algorithmic primitive that plays an important role in context of both variational quantum algorithms (VQA) and fully fault-tolerant ones, and has been applied to a diverse range of problems. Conventionally, Pauli basis is used for LCU decomposition, which however in worst case can result in number of LCU terms that scale quadratically with respect to the matrix size. We show that by using an alternate basis one can better exploit the sparsity and underlying structure of matrix leading to number of tensor product terms which scale only logarithmically with respect to the matrix size. We develop numerical and semi-analytical approaches for computing sigma basis decomposition for an arbitrary matrix. Given this new basis is comprised of non-unitary operators, we employ the concept of unitary completion to design efficient quantum circuits for evaluation of the expectation values of operators composed of tensor product of elements from sigma basis which can be used for cost function evaluation in VQAs. We also develop an approach for block encoding of arbitrary operator given its decomposition in sigma basis which could be used in variety of fully fault-tolerant algorithms. We compare our approach with other related concepts in the literature including unitary dilation and provide numerical illustrations on several PDE examples.
Worldwide visual geo-localization seeks to determine the geographic location of an image anywhere on Earth using only its visual content. Learned representations of geography for visual geo-localization remain an active research topic despite much progress. We formulate geo-localization as aligning the visual representation of the query image with a learned geographic representation. Unlike prior work, our geographic representation explicitly models the world as a hierarchy of learned geographic embeddings. Additionally, we introduce an approach to efficiently fuse the appearance features of the query image with its semantic segmentation map, forming a robust visual representation. Our main experiments demonstrate improved all-time bests in 22 out of 25 metrics measured across five benchmark datasets compared to prior state-of-the-art (SOTA) methods. Additional ablation studies support the claim that these gains are primarily driven by the combination of geographic and visual representations.
Unsupervised Domain Adaptation (UDA) is essential for enabling semantic segmentation in new domains without requiring costly pixel-wise annotations. State-of-the-art (SOTA) UDA methods primarily use self-training with architecturally identical teacher and student networks, relying on Exponential Moving Average (EMA) updates. However, these approaches face substantial performance degradation with lightweight models due to inherent architectural inflexibility leading to low-quality pseudo-labels. To address this, we propose Distilled Unsupervised Domain Adaptation (DUDA), a novel framework that combines EMA-based self-training with knowledge distillation (KD). Our method employs an auxiliary student network to bridge the architectural gap between heavyweight and lightweight models for EMA-based updates, resulting in improved pseudo-label quality. DUDA employs a strategic fusion of UDA and KD, incorporating innovative elements such as gradual distillation from large to small networks, inconsistency loss prioritizing poorly adapted classes, and learning with multiple teachers. Extensive experiments across four UDA benchmarks demonstrate DUDA's superiority in achieving SOTA performance with lightweight models, often surpassing the performance of heavyweight models from other approaches.
Combinatorial optimization problems such as Maximum Independent Set (MIS) and Maximum Cut (MaxCut) arise across a wide range of applications, yet solving large-scale instances remains computationally challenging. While discrete local-search procedures can produce high-quality solutions, their inherently sequential and iterative nature can become increasingly expensive as problem size and density grow. In this paper, we present a differentiable perspective on combinatorial optimization that leverages continuous relaxation with parallelizable gradient-based updates to improve scalability while reducing dependence on discrete local search. More specifically, for MIS, we show that solutions that remain repairable through a discrete (1,2)-swap can nevertheless be fixed points of projected gradient ascent, revealing a fundamental limitation of purely local gradient dynamics. For MaxCut, we develop a new quadratic formulation whose landscape provably allows projected gradient ascent to escape 1-flip-repairable solutions while ensuring binary local maximizers. Building on these insights, and inspired by discrete perturb-and-improve methods, we introduce a framework that combines projected gradient ascent with differentiable mutations (partial resets) to efficiently explore new regions of the solution space}. Experimental results on MIS and MaxCut demonstrate the potential of this approach for scalable combinatorial optimization, particularly in regimes where conventional discrete local search becomes computationally prohibitive.
Generative artificial intelligence (AI) and natural language processing (NLP) applications, particularly those utilizing large language models (LLMs) with Retrieval-Augmented Generation (RAG), fine-tuning and their variations, are receiving widespread attention. However, despite their promise, the dependability of LLM applications remains uncertain. This study introduces a new dimension to improving LLM performance through systematic query classification. A taxonomy-based approach was developed to categorize queries by their likelihood of eliciting accurate responses, from those likely to succeed to those prone to failure. Bloom's Taxonomy (BT) provided the conceptual direction for assessing and classifying queries using a curated dataset of question-answer pairs mapped to BT levels to evaluate RAG-LLM performance across categories. Building on insights from BT-based analysis, an innovative 'Taxonomical Query Classifier' (TQC) framework for LLMs is introduced: Finder (F), Evaluator (Ev), Explainer (Ex), and Generator (G)-collectively termed FEEG. FEEG-TQC classifies queries to improve predictability, accuracy, and enhance human-AI interaction. It differentiates queries based on intent, whether they (F) seek factual information, (Ev) evaluate content, (Ex) explain concepts, or (G) generate new content. By developing the concept of query category variations, this research provides a systematic approach to managing query quality and predictive estimations of LLM response accuracy. The findings contribute to the development of more effective human-AI interaction opportunities for LLM-based generative AI systems, emphasizing query classification as an important factor in improving generative AI systems.
