We propose a multi-tier paradigm to preserve various components of Morse-Smale complexes in lossy compressed scalar fields, including extrema, saddles, separatrices, and persistence diagrams. Existing error-bounded lossy compressors rarely consider preserving topological structures such as discrete Morse-Smale complexes, leading to significant inaccuracies in data interpretation and potentially resulting in incorrect scientific conclusions. This paper mainly focuses on preserving the Morse-Smale complexes in 2D or 3D discrete scalar fields by precisely preserving critical simplices and the separatrices that connect them. Our approach generates a series of edits during compression time, which are applied to the decompressed data to accurately reconstruct the complexes while maintaining the error within prescribed bounds. We design a workflow that iteratively fixes critical simplices and separatrices in alternating steps until convergence within finite iterations. Our approach addresses diverse application needs by offering users flexible options to balance compression efficiency and feature preservation. To enable effective integration with lossy compressors, we use GPU parallelism to enhance the performance of each workflow component. We conduct experiments on various datasets to demonstrate the effectiveness of our method in accurately preserving Morse-Smale complexes.
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Fine-tuning Large Language Models (LLMs) incurs considerable training costs, driving the need for data-efficient training with optimised data ordering. Human-inspired strategies offer a solution by organising data based on human learning practices. This study evaluates the fine-tuning efficiency of five human-inspired strategies across four language models, three datasets, and both human- and LLM-labelled data in the context of medical question answering. These strategies achieve the best accuracy gain of 1.81% and an average gain of 1.02% across datasets, with interleaved strategies delivering the best average results. However, the best strategy varies across model-dataset combinations, limiting the generalisability of the effects of any single strategy. Additionally, LLM-defined question difficulty outperforms human-defined labels in curriculum-based learning, showing the potential of model-generated data as a cost-effective alternative for optimising fine-tuning.
The Traveling Salesman Problem (TSP) in the two-dimensional Euclidean plane is among the oldest and most famous NP-hard optimization problems. In breakthrough works, Arora [J. ACM 1998] and Mitchell [SICOMP 1999] gave the first polynomial time approximation schemes. The running time of their approximation schemes was improved by Rao and Smith [STOC 1998] to $(1/\varepsilon)^{O(1/\varepsilon)} n \log n$. Bartal and Gottlieb [FOCS 2013] gave an approximation scheme of running time $2^{(1/\varepsilon)^{O(1)}} n$, which is optimal in $n$. Recently, Kisfaludi-Bak, Nederlof, and W\k{e}grzycki [FOCS 2021] gave a $2^{O(1/\varepsilon)} n \log n$ time approximation scheme, achieving the optimal running time in $\varepsilon$ under the Gap-ETH conjecture. In our work, we give a $2^{O(1/\varepsilon)} n$ time approximation scheme, achieving the optimal running time both in $n$ and in $\varepsilon$ under the Gap-ETH conjecture.
We construct a polynomial-time classical algorithm that samples from the output distribution of low-depth noisy Clifford circuits with any product-state inputs and final single-qubit measurements in any basis. This class of circuits includes Clifford-magic circuits and Conjugated-Clifford circuits, which are important candidates for demonstrating quantum advantage using non-universal gates. Additionally, our results generalize a simulation algorithm for IQP circuits [Rajakumar et. al, SODA'25] to the case of IQP circuits augmented with CNOT gates, which is another class of non-universal circuits that are relevant to current experiments. Importantly, our results do not require randomness assumptions over the circuit families considered (such as anticoncentration properties) and instead hold for \textit{every} circuit in each class. This allows us to place tight limitations on the robustness of these circuits to noise. In particular, we show that there is no quantum advantage at large depths with realistically noisy Clifford circuits, even with perfect magic state inputs, or IQP circuits with CNOT gates, even with arbitrary diagonal non-Clifford gates. The key insight behind the algorithm is that interspersed noise causes a decay of long-range entanglement, and at depths beyond a critical threshold, the noise builds up to an extent that most correlations can be classically simulated. To prove our results, we merge techniques from percolation theory with tools from Pauli path analysis.
Semantic segmentation models are typically trained on a fixed set of classes, limiting their applicability in open-world scenarios. Class-incremental semantic segmentation aims to update models with emerging new classes while preventing catastrophic forgetting of previously learned ones. However, existing methods impose strict rigidity on old classes, reducing their effectiveness in learning new incremental classes. In this work, we propose Taxonomy-Oriented Poincar\'e-regularized Incremental-Class Segmentation (TOPICS) that learns feature embeddings in hyperbolic space following explicit taxonomy-tree structures. This supervision provides plasticity for old classes, updating ancestors based on new classes while integrating new classes at fitting positions. Additionally, we maintain implicit class relational constraints on the geometric basis of the Poincar\'e ball. This ensures that the latent space can continuously adapt to new constraints while maintaining a robust structure to combat catastrophic forgetting. We also establish eight realistic incremental learning protocols for autonomous driving scenarios, where novel classes can originate from known classes or the background. Extensive evaluations of TOPICS on the Cityscapes and Mapillary Vistas 2.0 benchmarks demonstrate that it achieves state-of-the-art performance. We make the code and trained models publicly available at //topics.cs.uni-freiburg.de.
We study gradient descent (GD) dynamics on logistic regression problems with large, constant step sizes. For linearly-separable data, it is known that GD converges to the minimizer with arbitrarily large step sizes, a property which no longer holds when the problem is not separable. In fact, the behaviour can be much more complex -- a sequence of period-doubling bifurcations begins at the critical step size $2/\lambda$, where $\lambda$ is the largest eigenvalue of the Hessian at the solution. Using a smaller-than-critical step size guarantees convergence if initialized nearby the solution: but does this suffice globally? In one dimension, we show that a step size less than $1/\lambda$ suffices for global convergence. However, for all step sizes between $1/\lambda$ and the critical step size $2/\lambda$, one can construct a dataset such that GD converges to a stable cycle. In higher dimensions, this is actually possible even for step sizes less than $1/\lambda$. Our results show that although local convergence is guaranteed for all step sizes less than the critical step size, global convergence is not, and GD may instead converge to a cycle depending on the initialization.
Current speech-based LLMs are predominantly trained on extensive ASR and TTS datasets, excelling in tasks related to these domains. However, their ability to handle direct speech-to-speech conversations remains notably constrained. These models often rely on an ASR-to-TTS chain-of-thought pipeline, converting speech into text for processing before generating audio responses, which introduces latency and loses audio features. We propose a method that implicitly internalizes ASR chain of thought into a speech LLM, enhancing its native speech understanding capabilities. Our approach reduces latency and improves the model's native understanding of speech, paving the way for more efficient and natural real-time audio interactions. We also release a large-scale synthetic conversational dataset to facilitate further research.
Feature selection using Wasserstein Distance and application to TCGA-LUAD multi-omics data
We show through numerical simulation that the Quantum Approximate Optimization Algorithm (QAOA) for higher-order, random-coefficient, heavy-hex compatible spin glass Ising models has strong parameter concentration across problem sizes from $16$ up to $127$ qubits for $p=1$ up to $p=5$, which allows for straight-forward transfer learning of QAOA angles on instance sizes where exhaustive grid-search is prohibitive even for $p>1$. We use Matrix Product State (MPS) simulation at different bond dimensions to obtain confidence in these results, and we obtain the optimal solutions to these combinatorial optimization problems using CPLEX. In order to assess the ability of current noisy quantum hardware to exploit such parameter concentration, we execute short-depth QAOA circuits (with a CNOT depth of 6 per $p$, resulting in circuits which contain $1420$ two qubit gates for $127$ qubit $p=5$ QAOA) on $100$ higher-order (cubic term) Ising models on IBM quantum superconducting processors with $16, 27, 127$ qubits using QAOA angles learned from a single $16$-qubit instance. We show that (i) the best quantum processors generally find lower energy solutions up to $p=3$ for 27 qubit systems and up to $p=2$ for 127 qubit systems and are overcome by noise at higher values of $p$, (ii) the best quantum processors find mean energies that are about a factor of two off from the noise-free numerical simulation results. Additional insights from our experiments are that large performance differences exist among different quantum processors even of the same generation and that dynamical decoupling significantly improve performance for some, but decrease performance for other quantum processors. Lastly we show $p=1$ QAOA angle mean energy landscapes computed using up to a $414$ qubit quantum computer, showing that the mean QAOA energy landscapes remain very similar as the problem size changes.
We describe a mesh-free three-dimensional (3D) numerical scheme for solving the incompressible semi-geostrophic equations, based on semi-discrete optimal transport techniques. These results generalise previous two-dimensional (2D) implementations. The optimal transport methods we adopt are known for their structural preservation and energy conservation qualities and achieve an excellent level of efficiency and numerical energy-conservation. We use this scheme to generate numerical simulations of an important benchmark problem. To our knowledge, this is the first fully 3D simulation of this particular cyclone, evidencing the model's applicability to atmospheric and oceanic phenomena and offering a novel, robust tool for meteorological and oceanographic modelling.
Structured state-space models (SSMs) such as S4, stemming from the seminal work of Gu et al., are gaining popularity as effective approaches for modeling sequential data. Deep SSMs demonstrate outstanding performance across a diverse set of domains, at a reduced training and inference cost compared to attention-based transformers. Recent developments show that if the linear recurrence powering SSMs allows for multiplicative interactions between inputs and hidden states (e.g. GateLoop, Mamba, GLA), then the resulting architecture can surpass in both in accuracy and efficiency attention-powered foundation models trained on text, at scales of billion parameters. In this paper, we give theoretical grounding to this recent finding using tools from Rough Path Theory: we show that when random linear recurrences are equipped with simple input-controlled transitions (selectivity mechanism), then the hidden state is provably a low-dimensional projection of a powerful mathematical object called the signature of the input -- capturing non-linear interactions between tokens at distinct timescales. Our theory not only motivates the success of modern selective state-space models such as Mamba but also provides a solid framework to understand the expressive power of future SSM variants.
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