The Sign-Symmetric Hassanat Distance (SHasD), a single branch-free formula D(a,b)=|a−b|/(1+max(|a|,|b|)) that is invariant under the reflection x↦−x, coincides exactly with HasD on non-negative data, and removes the conditional shift entirely.
Abstract
The Hassanat Distance (HasD) is a bounded, non-convex metric widely used in k-nearest-neighbor (KNN) classification for its robustness to noise, outliers, and heterogeneous feature scales. Its definition, however, breaks a natural symmetry: through a sign-dependent shift it assigns different distances to mirror-image pairs such as (1,2) and (−1,−2), distorting neighborhoods exactly in the value ranges that modern preprocessing (z-scoring, principal component analysis (PCA), learned embeddings) produces. We introduce the Sign-Symmetric Hassanat Distance (SHasD), a single branch-free formula D(a,b)=|a−b|/(1+max(|a|,|b|)) that is invariant under the reflection x↦−x, coincides exactly with HasD on non-negative data, and removes the conditional shift entirely. We prove SHasD is a metric, and we derive a range-normalized companion, SHasD-R, that additionally restores ray monotonicity and the [0,1) per-dimension bound. On 23 datasets across three normalization regimes and ten distance measures, SHasD improves significantly on HasD on data containing negative values (mean gain +1.1 percentage points, up to +7.4; Wilcoxon p=0.0026, Holm-corrected) and attains the best mean rank of the compared measures on signed, heavy-tailed, outlier-rich data, while preserving HasD’s robustness. An additive per-dimension decomposition yields a built-in interpretation of every prediction.
It is shown in a variety of image classification settings and on several datasets, that quasibinary classifiers are considerably better in classification settings where regular binary and softmax classifiers suffer, including zero-label and multi-label classification.
Shuai Liao, E. Gavves, Changyong Oh et al.· 0 citations
Global Covariance Pooling (GCP) improves deep networks by capturing second-order feature statistics, and is especially effective for fine-grained recognition. Because covariance matrices live on the Symmetric Positive Definite (SPD) manifold, a normalization step is required before the Euclidean classifier. The faithful choice is the matrix logarithm (MLN-COV), which maps the SPD manifold to its tangent space; in practice it was abandoned in favour of the matrix square root because its eigendecomposition-based gradient is numerically unstable. We show that this instability is an artifact of computing the logarithm spectrally, not of the logarithm itself. Approximating the logarithm with finite polynomials in the covariance matrix removes the eigendecomposition from both passes: every operation becomes a General Matrix Multiplication (GEMM), the gradient stays bounded on the spectral support of the pre-normalized covariance, and the unstable 1/(lambda_i-lambda_j) term never appears. The key ingredient is a mean-eigenvalue pre-normalization that centres the spectrum near 1, away from the singularity of log, with a scalar post-compensation that returns the singular part of log(A) in closed form. Our recommended normalizer is a degree-8 Chebyshev expansion evaluated by a three-term matrix recurrence, with a matching reverse recurrence for the backward pass; Legendre, Laguerre, Taylor and Pade expansions are studied as controls that isolate the roles of the basis and of the target function. On three fine-grained benchmarks and ImageNet-1k the decomposition-free logarithm is both faster and more accurate than the spectral logarithm and than the square-root approximations it replaces, and at matched basis and degree the log target beats the square-root target, confirming that the gain comes from the faithful Riemannian map rather than from a better polynomial family.
Md Rifat Ur Rahman, Md Raihan Khan, Md Sakib Hossain Shovon et al.· 0 citations
A membership-restricted Shepard rho resolves single-point and minority-population questions that many-pair statistics cannot -- questions on which even DREAMS, a recent local-plus-global hybrid, fails silently.
We introduce the Directional Kernel Mean Difference (DKMD), a signed statistic for univariate distribution comparison that preserves the direction of distributional shifts. Unlike the squared Maximum Mean Discrepancy (MMD), which discards directional information by squaring the RKHS distance, DKMD integrates the difference of kernel mean embeddings against a fixed odd weighting function. This construction yields three structural properties: antisymmetry, immunity to symmetric distributional differences, and directional monotonicity under stochastic dominance. We derive a data-driven Riemann estimator that ensures asymptotic consistency with the continuous formulation, strictly preserving the theoretical guarantees of the signed statistic in empirical evaluations. To overcome the quadratic computational cost of kernel methods, we develop an $O(N \log N)$ prefix--suffix scanning algorithm that exploits the total order of the real line while requiring only $O(N)$ memory. Experiments on synthetic benchmarks demonstrate that DKMD correctly isolates directional shifts from symmetric perturbations, remains robust to heavy-tailed outliers that can flip the sign of the mean difference, and scales to millions of samples in seconds.
In real-world scenarios, the training data usually contains redundant features, label noise and feature noise, which provide severe challenges for the efficiency of machine learning methods. Since standard support vector machine (SVM) adopts $l_2$-norm penalty and hinge loss function, it lacks the ability of selecting significant features and is sensitive to noise. To address these issues, this paper proposes a novel asymmetric, robust, bounded, sparse and smooth (aR) loss function for $l_1$-norm penalized geometric twin SVM (aRSGTSVM) to handle classification and regression tasks. The $l_1$-norm penalty can achieve the feature selection. The proposed aR loss function can not only effectively mitigate the impact of label noise, but also significantly enhance the stability to resampling noise, i.e., the zero-mean feature noise around the boundary hyperplanes. Furthermore, a statistical analysis of the robustness of aRSGTSVM was also conducted using the influence function. Since aRSGTSVM involves nonconvex and nonsmooth optimization, we develop a fast and stable proximal gradient descent based solving algorithm. Compared with related state-of-the-art methods, experimental results demonstrate the superiority of the proposed aRSGTSVM on both synthetic and UCI datasets. Furthermore, we apply aRSGTSVM to index tracking tasks, where results for tracking the different indices in the China stock market show that it can achieve satisfactory performance.
The pre-softmax score of an attention head is a bilinear form $score(i,j) = x_i^T M x_j$ in a learned operator $M = W_q^T W_k$. Because M is generally non-symmetric, hence non-normal, it has a complex eigenspectrum and non-orthogonal eigenvectors, the regime where non-Hermitian and random-matrix tools apply. We ask what this spectrum encodes, at three levels for previous-token and induction circuits. Statically, across seven pretrained models spanning three positional schemes, the strongest previous-token heads are spectrally rotational under RoPE and non-rotational, or content-like, where position enters outside QK (learned-absolute and ALiBi); the model-level separation is perfect at every top-k examined (exact permutation $p=0.029$), and zeroing the per-frequency RoPE phase $Im(M_t)$ eliminates induction on a pre-identified previous-token head in all three RoPE models. Dynamically, over public Pythia checkpoints every head originates at the random-matrix (Ginibre) null; the rotational signature emerges with the behavior, not before it, and the population-median suppression that yields the final profile follows circuit formation, so the profile is a consolidated fingerprint, not a precursor. Causally, and at toy scale, no spectral channel is necessary: constrained two-layer training reroutes around every ban with capability intact, albeit at a significant formation delay (four pre-registered contrasts, $q_BH<= 0.016$). The cost structure exposes each scheme's default: imposing symmetry slows learned-absolute models by a factor of 2.9, whereas a RoPE head with a fully symmetric static M still routes directionally via the phase channel, impossible under absolute positions. Within the settings examined, the positional scheme sets the default spectral algebra of an attention head's solution: a fingerprint sculpted after function, not a hard constraint upon it.
