However, the competitive performance by leveraging 3D networks results in huge computational costs, which are far beyond that of 2D networks. In this paper, we propose a novel Hilbert curve-based cross-dimensionality distillation approach that facilitates the knowledge of 3D networks to improve the performance of 2D networks. The proposed Hilbert Distillation (HD) method preserves the structural information via the Hilbert curve, which maps high-dimensional (>=2) representations to one-dimensional continuous space-filling curves. Since the distilled 2D networks are supervised by the curves converted from dimensionally heterogeneous 3D features, the 2D networks are given an informative view in terms of learning structural information embedded in well-trained high-dimensional representations. We further propose a Variable-length Hilbert Distillation (VHD) method to dynamically shorten the walking stride of the Hilbert curve in activation feature areas and lengthen the stride in context feature areas, forcing the 2D networks to pay more attention to learning from activation features. The proposed algorithm outperforms the current state-of-the-art distillation techniques adapted to cross-dimensionality distillation on two classification tasks. Moreover, the distilled 2D networks by the proposed method achieve competitive performance with the original 3D networks, indicating the lightweight distilled 2D networks could potentially be the substitution of cumbersome 3D networks in the real-world scenario.

Sequence models such as transformers require inputs to be represented as one-dimensional sequences. In vision, this typically involves flattening images using a fixed row-major (raster-scan) order. While full self-attention is permutation-equivariant, modern long-sequence transformers increasingly rely on architectural approximations that break this invariance and introduce sensitivity to patch ordering. We show that patch order significantly affects model performance in such settings, with simple alternatives like column-major or Hilbert curves yielding notable accuracy shifts. Motivated by this, we propose REOrder, a two-stage framework for discovering task-optimal patch orderings. First, we derive an information-theoretic prior by evaluating the compressibility of various patch sequences. Then, we learn a policy over permutations by optimizing a Plackett-Luce policy using REINFORCE. This approach enables efficient learning in a combinatorial permutation space. REOrder improves top-1 accuracy over row-major ordering on ImageNet-1K by up to 3.01% and Functional Map of the World by 13.35%.

Designing sparse attention for diffusion transformers requires reconciling two-dimensional spatial locality with GPU efficiency, a trade-off that current methods struggle to achieve. Existing approaches enforce two-dimensional spatial locality but often incur uncoalesced memory access. We present HilbertA, a 2D-aware and GPU-efficient sparse attention mechanism. HilbertA reorders image tokens along Hilbert curves to achieve a contiguous memory layout while preserving spatial neighborhoods, and employs a sliding schedule across layers to enable long-range information propagation without repeated or uncoalesced memory access. To further enhance cross-tile communication and positional awareness, HilbertA introduces a small central shared region. Implemented in Triton, HilbertA delivers comparable image quality with significant acceleration over prior methods on Flux.1-dev, demonstrating the feasibility of hardware-aligned two-dimensional sparse attention for high-resolution image generation. HilbertA delivers attention speedups of $2.3\times$ when generating $1024\times 1024$ images, and up to $4.17\times$ at $2048\times 2048$, while achieving image quality comparable to or surpassing baselines.

Draw a random matrix according to the construction specified in section 3.2 and fix it.

Accurate molecular sequence analysis is a key task in the field of bioinformatics. To apply molecular sequence classification algorithms, we first need to generate the appropriate representations of the sequences. Traditional numeric sequence representation techniques are mostly based on sequence alignment that faces limitations in the form of lack of accuracy. Although several alignment-free techniques have also been introduced, their tabular data form results in low performance when used with Deep Learning (DL) models compared to the competitive performance observed in the case of image-based data. To find a solution to this problem and to make Deep Learning (DL) models function to their maximum potential while capturing the important spatial information in the sequence data, we propose a universal Hibert curve-based Chaos Game Representation (CGR) method. This method is a transformative function that involves a novel Alphabetic index mapping technique used in constructing Hilbert curve-based image representation from molecular sequences. Our method can be globally applied to any type of molecular sequence data. The Hilbert curve-based image representations can be used as input to sophisticated vision DL models for sequence classification. The proposed method shows promising results as it outperforms current state-of-the-art methods by achieving a high accuracy of $94.5$\% and an F1 score of $93.9\%$ when tested with the CNN model on the lung cancer dataset. This approach opens up a new horizon for exploring molecular sequence analysis using image classification methods.

However, the competitive performance by leveraging 3D networks results in huge computational costs, which are far beyond that of 2D networks. In this paper, we propose a novel Hilbert curve-based cross-dimensionality distillation approach that facilitates the knowledge of 3D networks to improve the performance of 2D networks. The proposed Hilbert Distillation (HD) method preserves the structural information via the Hilbert curve, which maps high-dimensional ( 2) representations to one-dimensional continuous space-filling curves. Since the distilled 2D networks are supervised by the curves converted from dimensionally heterogeneous 3D features, the 2D networks are given an informative view in terms of learning structural information embedded in well-trained high-dimensional representations. We further propose a Variable-length Hilbert Distillation (VHD) method to dynamically shorten the walking stride of the Hilbert curve in activation feature areas and lengthen the stride in context feature areas, forcing the 2D networks to pay more attention to learning from activation features.

Landscape analysis aims to characterise optimisation problems based on their objective (or fitness) function landscape properties. The problem search space is typically sampled, and various landscape features are estimated based on the samples. One particularly salient set of features is information content, which requires the samples to be sequences of neighbouring solutions, such that the local relationships between consecutive sample points are preserved. Generating such spatially correlated samples that also provide good search space coverage is challenging. It is therefore common to first obtain an unordered sample with good search space coverage, and then apply an ordering algorithm such as the nearest neighbour to minimise the distance between consecutive points in the sample. However, the nearest neighbour algorithm becomes computationally prohibitive in higher dimensions, thus there is a need for more efficient alternatives. In this study, Hilbert space-filling curves are proposed as a method to efficiently obtain high-quality ordered samples. Hilbert curves are a special case of fractal curves, and guarantee uniform coverage of a bounded search space while providing a spatially correlated sample. We study the effectiveness of Hilbert curves as samplers, and discover that they are capable of extracting salient features at a fraction of the computational cost compared to Latin hypercube sampling with post-factum ordering. Further, we investigate the use of Hilbert curves as an ordering strategy, and find that they order the sample significantly faster than the nearest neighbour ordering, without sacrificing the saliency of the extracted features.

Quantization is widely used to increase deep neural networks' (DNN) memory, computation, and power efficiency. Various techniques, such as post-training quantization and quantization-aware training, have been proposed to improve quantization quality. We introduce a novel approach for DNN quantization that uses a redundant representation of DNN's output. We represent the target quantity as a point on a 2D parametric curve. The DNN model is modified to predict 2D points that are mapped back to the target quantity at a post-processing stage. We demonstrate that this mapping can reduce quantization error. For the low-order parametric Hilbert curve, Depth-From-Stereo task, and two models represented by U-Net architecture and vision transformer, we achieved a quantization error reduction by about 5 times for the INT8 model at both CPU and DSP delegates. This gain comes with a minimal inference time increase (less than 7%). Our approach can be applied to other tasks, including segmentation, object detection, and key-points prediction.