arXiv subjects

Pablo Acuaviva

Publications and source records attributed to Pablo Acuaviva.

6 recordsLinked to original sources

arXiv2026

CNOT-Distance is NP-complete under all-to-all connectivity

Given $A\in\operatorname{GL}(N,2)$ and an integer $K$, we ask whether $A$ can be implemented by at most $K$ CNOT gates on fixed labelled wires with all-to-all connectivity. We prove that this problem is NP-complete. From a finite simple graph $G=(V,E)$, we construct an upper-unitriangular matrix $A_G\in\operatorname{GL}(2|V|+|E|+1,2)$ satisfying $\ell_{\mathrm{CNOT}}(A_G)=2|V|+2|E|+\tau(G)$, where $\tau(G)$ is the minimum vertex-cover size. Each target matrix has $O(N)$ nonzero entries and row Hamming weight at most four. The lower bound unfolds an arbitrary CNOT circuit into an XOR directed acyclic graph and applies projection--contraction operations, allowing cancellation and unrestricted reuse of intermediate parities. For this family, the optimum is unchanged by any finite number of clean or borrowed ancillary wires that must be restored. A polynomial-time decoder further yields NP-hardness of approximation within every fixed additive constant and, through an L-reduction from Minimum Vertex Cover on cubic graphs, APX-hardness of the associated CNOT-circuit optimisation problem.

quant-ph↗

arXiv2026

Mathematical Discovery in the Wild: AI-Guided Proofs in Banach Space Theory

We investigate the capacity of current language models to contribute to mathematical research. In Banach space theory, AI systems generated key ideas and proofs for five new results, which were then verified and refined by humans. We also developed an automated system that searches the literature for open problems and attempts solutions at scale. Our results show both the potential of language models for mathematical discovery and the continuing importance of expert verification.

math.FA↗

arXiv2026

A Unital Banach Algebra Which Is Not a Calkin Algebra

Let $\mathscr{A}(X)$ and $\mathscr{K}(X)$ denote the ideals of approximable and compact operators on a Banach space $X$, respectively. We construct a unital Banach algebra $A$ of density character $\mathfrak c=2^{\aleph_0}$, with exactly one non-zero proper closed two-sided ideal, such that, for every Banach space $X$, the algebra $A$ is isomorphic to neither $\mathscr{B}(X)/\mathscr{A}(X)$ nor $\mathscr{B}(X)/\mathscr{K}(X)$. Thus $A$ is not a Calkin algebra under either of the two customary conventions.

math.FA↗

arXiv2026

Communication-Inspired Tokenization for Structured Image Representations

Discrete image tokenizers have emerged as a key component of modern vision and multimodal systems, providing a sequential interface for transformer-based architectures. However, most existing approaches remain primarily optimized for reconstruction and compression, often yielding tokens that capture local texture rather than object-level semantic structure. Inspired by the incremental and compositional nature of human communication, we introduce COMmunication inspired Tokenization (COMiT), a framework for learning structured discrete visual token sequences. COMiT constructs a latent message within a fixed token budget by iteratively observing localized image crops and recurrently updating its discrete representation. At each step, the model integrates new visual information while refining and reorganizing the existing token sequence. After several encoding iterations, the final message conditions a flow-matching decoder that reconstructs the full image. Both encoding and decoding are implemented within a single transformer model and trained end-to-end using a combination of flow-matching reconstruction and semantic representation alignment losses. Our experiments demonstrate that while semantic alignment provides grounding, attentive sequential tokenization is critical for inducing interpretable, object-centric token structure and substantially improving compositional generalization and relational reasoning over prior methods.

cs.CV↗

arXiv2025

Rethinking Visual Intelligence: Insights from Video Pretraining

Large language models (LLMs) have demonstrated that large-scale pretraining enables systems to adapt rapidly to new problems with little supervision in the language domain. This success, however, has not translated as effectively to the visual domain, where models, including LLMs, continue to struggle with compositional understanding, sample efficiency, and general-purpose problem-solving. We investigate Video Diffusion Models (VDMs) as a promising direction for bridging this gap. Pretraining on spatiotemporal data endows these models with strong inductive biases for structure and dynamics, which we hypothesize can support broad task adaptability. To test this, we design a controlled evaluation in which both a pretrained LLM and a pretrained VDM are equipped with lightweight adapters and presented with tasks in their natural modalities. Across benchmarks including ARC-AGI, ConceptARC, visual games, route planning, and cellular automata, VDMs demonstrate higher data efficiency than their language counterparts. Taken together, our results indicate that video pretraining offers inductive biases that support progress toward visual foundation models.

cs.CV↗

arXiv2025

From Generation to Generalization: Emergent Few-Shot Learning in Video Diffusion Models

Video Diffusion Models (VDMs) have emerged as powerful generative tools, capable of synthesizing high-quality spatiotemporal content. Yet, their potential goes far beyond mere video generation. We argue that the training dynamics of VDMs, driven by the need to model coherent sequences, naturally pushes them to internalize structured representations and an implicit understanding of the visual world. To probe the extent of this internal knowledge, we introduce a few-shot fine-tuning framework that repurposes VDMs for new tasks using only a handful of examples. Our method transforms each task into a visual transition, enabling the training of LoRA weights on short input-output sequences without altering the generative interface of a frozen VDM. Despite minimal supervision, the model exhibits strong generalization across diverse tasks, from low-level vision (for example, segmentation and pose estimation) to high-level reasoning (for example, on ARC-AGI). These results reframe VDMs as more than generative engines. They are adaptable visual learners with the potential to serve as the backbone for future foundation models in vision.

cs.CV↗