GARLIC: GAussian Representation LearnIng for spaCe partitioning

TL;DR

The paper addresses efficient high-dimensional Euclidean ANN search by learning geometry-aware space partitions. It introduces GARLIC, which represents $\mathbb{R}^d$ with Gaussian cells that are anisotropic and density-adaptive, aligning with local principal directions and refining capacity locally via split/clone operations. The learning objective is information-theoretic, balancing expected coverage, overlap, and probe-budget efficiency, with query-time processing using Mahalanobis distance to select cells and prune candidates via localized quantization. Empirical results on standard benchmarks show competitive recall–efficiency trade-offs, robustness to training on a small subset of data, and provide complexity analyses for build, query, and storage. Overall GARLIC demonstrates a principled, geometry-aware approach to ANN that improves locality and efficiency over isotropic or fixed-resolution partitions.

Abstract

We present \textbf{GARLIC}, a representation learning approach for Euclidean approximate nearest neighbor (ANN) search in high dimensions. Existing partitions tend to rely on isotropic cells, fixed global resolution, or balanced constraints, which fragment dense regions and merge unrelated points in sparse ones, thereby increasing the candidate count when probing only a few cells. Our method instead partitions \(\mathbb{R}^d\) into anisotropic Gaussian cells whose shapes align with local geometry and sizes adapt to data density. Information-theoretic objectives balance coverage, overlap, and geometric alignment, while split/clone refinement introduces Gaussians only where needed. At query time, Mahalanobis distance selects relevant cells and localized quantization prunes candidates. This yields partitions that reduce cross-cell neighbor splits and candidate counts under small probe budgets, while remaining robust even when trained on only a small fraction of the dataset. Overall, GARLIC introduces a geometry-aware space-partitioning paradigm that combines information-theoretic objectives with adaptive density refinement, offering competitive recall--efficiency trade-offs for Euclidean ANN search.

Figures (6)

• Figure 1: Additional distortion-aware accuracy results (higher is better). Each curve reports performance as a function of the candidate budget (x-axis). Panels show $\varepsilon$-Recall at $\varepsilon\in\{0.01,0.10\}$ and P@10 across SIFT1M, MNIST, and FMNIST. Methods closer to the top-left ($\nwarrow$) are more accurate under smaller candidate budgets.
• Figure 2: Approximation tightness summaries (lower is better). We report the 95th and 99th percentiles of the distance ratio ($r95$, $r99$) and the mean relative error ($RE_{\text{mean}} = \mathbb{E}[r-1]$) versus the candidate budget. Methods closer to the bottom-left ($\swarrow$) achieve tighter approximations with fewer candidates.
• Figure 3: Parameter ablations: (a) Impact of different loss term combinations on the Recall-Probe tradeoff, showing that the full loss ($L_{div}+L_{cov}+L_{anchor}$) provides the best balance. (b) Effect of split and clone operations, demonstrating that these operations improve efficiency while maintaining accuracy. (c) Impact of ReLU parameter $\tau$ in the divergence loss. (d) Effect of embedding dimensionality, showing GARLIC's robustness across different dimensions. (e) Efficient search via partial cell scanning. The blue dashed method regards the parameters used in main experiments, and top-left is better ($\nwarrow$).
• Figure 4: Diagnostic visualizations across Fashion-MNIST (left), MNIST (middle), and SIFT (right). Top row: minimum Mahalanobis reconstruction error; middle row: average Gaussian coverage per point; bottom row: relationship between local density and Gaussian coverage. Bottom row's colormap depicts frequency-density, in logarithmic scale.
• Figure 5: Curvature diagnostics across Fashion-MNIST (left), MNIST (middle), and SIFT (right). Row 1: standard deviation of local curvature vs radius; Row 2: mean local curvature vs radius; Row 3: Gaussian coverage vs curvature std; Row 4: Gaussian coverage vs curvature mean. Coloring accounts for density.