Asthefigureshows,any true positives or false positives are assigned with ranksi. From the above demonstration, if any swap happens in a rank list between true and false positives,the mis-rank ofthat true positiveisdefinitely changed and will only result inincrease or decreaseofmandiby1. To determine whether the lower bound is tight is a little bit difficult. We firstly introduce some concepts and assumptions to make it easier. Let us start at the example placed in beginning of AppendixA.

Hash center-based deep hashing methods improve upon pairwise or triplet-based approaches by assigning fixed hash centers to each class as learning targets, thereby avoiding the inefficiency of local similarity optimization. However, random center initialization often disregards inter-class semantic relationships. While existing two-stage methods mitigate this by first refining hash centers with semantics and then training the hash function, they introduce additional complexity, computational overhead, and suboptimal performance due to stage-wise discrepancies. To address these limitations, we propose $\textbf{Center-Reassigned Hashing (CRH)}$, an end-to-end framework that $\textbf{dynamically reassigns hash centers}$ from a preset codebook while jointly optimizing the hash function. Unlike previous methods, CRH adapts hash centers to the data distribution $\textbf{without explicit center optimization phases}$, enabling seamless integration of semantic relationships into the learning process. Furthermore, $\textbf{a multi-head mechanism}$ enhances the representational capacity of hash centers, capturing richer semantic structures. Extensive experiments on three benchmarks demonstrate that CRH learns semantically meaningful hash centers and outperforms state-of-the-art deep hashing methods in retrieval tasks.

Low-dimensional chaotic systems such as the Lorenz-63 model are commonly used to benchmark system-agnostic methods for learning dynamics from data. Here we show that learning from noise-free observations in such systems can be achieved up to machine precision: using ordinary least squares regression on high-degree polynomial features with 512-bit arithmetic, our method exceeds the accuracy of standard 64-bit numerical ODE solvers of the true underlying dynamical systems. Depending on the configuration, we obtain valid prediction times of 32 to 105 Lyapunov times for the Lorenz-63 system, dramatically outperforming prior work that reaches 13 Lyapunov times at most. We further validate our results on Thomas' Cyclically Symmetric Attractor, a non-polynomial chaotic system that is considerably more complex than the Lorenz-63 model, and show that similar results extend also to higher dimensions using the spatiotemporally chaotic Lorenz-96 model. Our findings suggest that learning low-dimensional chaotic systems from noise-free data is a solved problem.

Deep hashing is an effective approach for large-scale image retrieval. Current methods are typically classified by their supervision types: point-wise, pair-wise, and list-wise. Recent point-wise techniques (e.g., CSQ, MDS) have improved retrieval performance by pre-assigning a hash center to each class, enhancing the discriminability of hash codes across various datasets. However, these methods rely on data-independent algorithms to generate hash centers, which neglect the semantic relationships between classes and may degrade retrieval performance. This paper introduces the concept of semantic hash centers, building on the idea of traditional hash centers. We hypothesize that hash centers of semantically related classes should have closer Hamming distances, while those of unrelated classes should be more distant. To this end, we propose a three-stage framework, SHC, to generate hash codes that preserve semantic structure. First, we develop a classification network to identify semantic similarities between classes using a data-dependent similarity calculation that adapts to varying data distributions. Second, we introduce an optimization algorithm to generate semantic hash centers, preserving semantic relatedness while enforcing a minimum distance between centers to avoid excessively similar hash codes. Finally, a deep hashing network is trained using these semantic centers to convert images into binary hash codes. Experimental results on large-scale retrieval tasks across several public datasets show that SHC significantly improves retrieval performance. Specifically, SHC achieves average improvements of +7.26%, +7.62%, and +11.71% in MAP@100, MAP@1000, and MAP@ALL metrics, respectively, over state-of-the-art methods.

The analysis of the compression effects in generative adversarial networks (GANs) after training, i.e. without any fine-tuning, remains an unstudied, albeit important, topic with the increasing trend of their computation and memory requirements. While existing works discuss the difficulty of compressing GANs during training, requiring novel methods designed with the instability of GANs training in mind, we show that existing compression methods (namely clipping and quantization) may be directly applied to compress GANs post-training, without any additional changes. High compression levels may distort the generated set, likely leading to an increase of outliers that may negatively affect the overall assessment of existing k-nearest neighbor (KNN) based metrics. We propose two new precision and recall metrics based on locality-sensitive hashing (LSH), which, on top of increasing the outlier robustness, decrease the complexity of assessing an evaluation sample against $n$ reference samples from $O(n)$ to $O(\log(n))$, if using LSH and KNN, and to $O(1)$, if only applying LSH. We show that low-bit compression of several pre-trained GANs on multiple datasets induces a trade-off between precision and recall, retaining sample quality while sacrificing sample diversity.