Selected work

Representative Papers

Bernstein-Schur Kernels: Random Features by Sketched Modulation and Radial Randomization

Jun 08, 2026

This work addresses the lack of effective random feature methods for Bernstein–Schur kernels—products of finite-dimensional feature maps and completely monotone translation-invariant kernels—which fall outside the scope of Bochner’s theorem and are incompatible with polynomial sketching. To overcome this, the authors propose a dual randomization strategy: applying matrix sketching to compress the finite-dimensional modulation component and combining Bernstein–Widder scale sampling with Gaussian random Fourier features for the radial completely monotone part. This approach yields the first unified, unbiased random feature construction for this kernel class, preserving the exact limiting behavior of the modulation while providing an explicit variance expression and operator norm bounds based on intrinsic dimensionality. Crucially, it decouples sketching error from radial sampling error. Under theoretical guarantees, only \(D_m\) features—far fewer than the \(O(d^2)\) required for exact modulation—suffice for high-accuracy approximation; for the YAT kernel instance, single-scale, single-frequency sampling achieves variance optimality under a fixed radial budget.

0 citationsRead paper

A Universal Reproducing Kernel Hilbert Space from Polynomial Alignment and IMQ Distance

May 04, 2026

This work addresses the challenge that existing reproducing kernel Hilbert space (RKHS) kernels struggle to simultaneously achieve universality, characteristicness, and strict positive definiteness. To resolve this, the paper introduces a novel kernel, termed the Yat kernel, which innovatively incorporates a polynomial alignment mechanism into inverse multiquadric (IMQ)-type kernels. By combining polynomial numerators with IMQ-based distance denominators, the Yat kernel constructs a positive definite kernel sensitive to non-radial directions within shared input or weight spaces. Theoretical analysis demonstrates that the Yat kernel attains universality, characteristicness, and strict positive definiteness over any compact domain, admits finite center expansions, and—through its three bias-equipped atomic components—exactly reconstructs arbitrary IMQ atoms in a dimension-independent manner. Leveraging Mercer’s theorem, Loewner order comparisons, and Rademacher complexity analysis, the study derives explicit generalization error bounds, confirming the model’s strong expressivity and learning stability.

0 citationsRead paper

In Defense of Cosine Similarity: Normalization Eliminates the Gauge Freedom

Feb 22, 2026

This work addresses the concern that using cosine similarity with unnormalized embeddings in matrix factorization models leads to arbitrary results due to gauge freedom, casting doubt on its validity. Through theoretical and geometric analysis, the authors demonstrate that the core issue lies not in cosine similarity itself, but in the mismatch between the training objective and the similarity metric. When embeddings are constrained to the unit hypersphere, gauge freedom is entirely eliminated. Under this constraint, cosine distance becomes strictly equivalent to Euclidean distance: specifically, the cosine distance equals half the squared Euclidean distance, and both induce identical neighbor rankings. This equivalence provides a rigorous theoretical foundation for the principled use of cosine similarity with normalized embeddings.

0 citationsRead paper

Recent publications

Latest Papers

Bernstein-Schur Kernels: Random Features by Sketched Modulation and Radial Randomization

Jun 08, 2026

This work addresses the lack of effective random feature methods for Bernstein–Schur kernels—products of finite-dimensional feature maps and completely monotone translation-invariant kernels—which fall outside the scope of Bochner’s theorem and are incompatible with polynomial sketching. To overcome this, the authors propose a dual randomization strategy: applying matrix sketching to compress the finite-dimensional modulation component and combining Bernstein–Widder scale sampling with Gaussian random Fourier features for the radial completely monotone part. This approach yields the first unified, unbiased random feature construction for this kernel class, preserving the exact limiting behavior of the modulation while providing an explicit variance expression and operator norm bounds based on intrinsic dimensionality. Crucially, it decouples sketching error from radial sampling error. Under theoretical guarantees, only \(D_m\) features—far fewer than the \(O(d^2)\) required for exact modulation—suffice for high-accuracy approximation; for the YAT kernel instance, single-scale, single-frequency sampling achieves variance optimality under a fixed radial budget.

0 citationsRead paper

A Universal Reproducing Kernel Hilbert Space from Polynomial Alignment and IMQ Distance

May 04, 2026

This work addresses the challenge that existing reproducing kernel Hilbert space (RKHS) kernels struggle to simultaneously achieve universality, characteristicness, and strict positive definiteness. To resolve this, the paper introduces a novel kernel, termed the Yat kernel, which innovatively incorporates a polynomial alignment mechanism into inverse multiquadric (IMQ)-type kernels. By combining polynomial numerators with IMQ-based distance denominators, the Yat kernel constructs a positive definite kernel sensitive to non-radial directions within shared input or weight spaces. Theoretical analysis demonstrates that the Yat kernel attains universality, characteristicness, and strict positive definiteness over any compact domain, admits finite center expansions, and—through its three bias-equipped atomic components—exactly reconstructs arbitrary IMQ atoms in a dimension-independent manner. Leveraging Mercer’s theorem, Loewner order comparisons, and Rademacher complexity analysis, the study derives explicit generalization error bounds, confirming the model’s strong expressivity and learning stability.

0 citationsRead paper

In Defense of Cosine Similarity: Normalization Eliminates the Gauge Freedom

Feb 22, 2026

This work addresses the concern that using cosine similarity with unnormalized embeddings in matrix factorization models leads to arbitrary results due to gauge freedom, casting doubt on its validity. Through theoretical and geometric analysis, the authors demonstrate that the core issue lies not in cosine similarity itself, but in the mismatch between the training objective and the similarity metric. When embeddings are constrained to the unit hypersphere, gauge freedom is entirely eliminated. Under this constraint, cosine distance becomes strictly equivalent to Euclidean distance: specifically, the cosine distance equals half the squared Euclidean distance, and both induce identical neighbor rankings. This equivalence provides a rigorous theoretical foundation for the principled use of cosine similarity with normalized embeddings.

0 citationsRead paper