PLS in the Mirror of Self-Attention

The Large Language Model from Power Law Decoder Representations (PLDR-LLM) and its attention, Power Law Graph Attention (PLGA), replace the fixed bilinear form of scaled dot-product attention (SDPA) with a learned, input-generated bilinear operator $G_{LM}$, built from a positive tensor $A_{LM}$ by elementwise power laws. The architecture is fully specified, verified against pinned reference releases; claims are labeled theorem, conditional theorem, measurement, or conjecture. Unconditionally: PLGA contains SDPA exactly at $G_{LM}=I$; $A_{LM}$ and $A_P$ are strictly entrywise positive, with Perron-Frobenius structure on $A_{LM}$; the DAG regularizer has the NOTEARS walk-counting form and positivity obstructs exact acyclicity; and, under nonresonance (satisfied by standard rotary frequencies), a commutant criterion identifies which operators preserve relative-position dependence. An inference-collapse theorem: exact input invariance of deductive outputs collapses inference to generalized SDPA with a constant operator. Measured invariance: relative fluctuations of $10^{-6}$ and below; perturbation bounds quantify but do not certify cached inference; the assembled proxy misses the decoding margin. A conditional three-stage mechanism (rotary twirl, concentration, row-map contraction) is measured on a released checkpoint. Blockwise training and scoring under the global Gram are stated with explicit target exposure; on tested samples, block and sequential scoring select identical answers and agree on the published TruthfulQA probability-mass metric within $5\times 10^{-5}$ per item. Self-organized criticality enters as a phenomenological framework with an intrinsic order parameter; open claims become falsifiable conjectures. Selected proof cores are machine-checked in Lean 4.

A standard self-attention layer consists of two interacting circuits: the query-key circuit that governs attention allocation, and the output-value circuit that maps attended representations to predictions. Collapsed and factorized parameterizations of the query-key and output-value circuits lead to qualitatively different attention patterns. In particular, some parameterizations give sharper attention to task-relevant tokens, at a similar training loss. We analyze how the parameterizations of these circuits shape the parameter trajectories in single-layer self-attention models trained for next-token prediction. Through gradient-flow analysis, we show that factorization induces implicit rescaling of the two circuits' learning rates. We derive closed-form dynamics showing that output-value and query-key parameters move along a line, with relative speeds determined by their learning rates. Faster query-key learning relative to output-value learning thus produces sharper attention, as the model compensates for slower output-value learning by increasing attention mass on relevant tokens. Experiments show that differences in the relative learning rates of the two circuits govern attention concentration. This improves attention interpretability proxies while maintaining comparable predictive performance.

Improved Kernel Partial Least Squares (IKPLS) algorithms 1 and 2 are among the fastest PLS calibration algorithms. This article focuses on two shared steps, the computation of the $\mathbf{X}$ rotations, $\mathbf{R}$, and the $\mathbf{Y}$ loadings, $\mathbf{Q}$, and accelerates both. For $\mathbf{R}$, term-by-term accumulation is replaced by a direct evaluation strategy that requires the same number of multiplications but parallelizes better on modern hardware. For $\mathbf{Q}$, I identify - to the best of my knowledge, for the first time - equivalences showing that each $\mathbf{Y}$ loading is obtainable, up to explicitly derived constants, from quantities already computed earlier in the same iteration, and I exploit them in IKPLS to reduce the cost of each loading from $Θ\left(KM\right)$ to $Θ\left(M\right)$ operations whenever $M = 1$ or $2 \leq M < K$, with $K$ predictor variables (number of columns in $\mathbf{X}$) and $M$ response variables (number of columns in $\mathbf{Y}$). Both improvements provably yield exactly the same $\mathbf{W}$, $\mathbf{P}$, $\mathbf{Q}$, $\mathbf{R}$, and $\mathbf{T}$ as the original algorithms. Benchmarks with NumPy (CPU) and JAX (GPU) show speedups of up to two orders of magnitude for the isolated steps and of approximately $2\times$ (CPU) and $6\times$ (GPU) for entire fits. Both improvements are implemented in the free, open-source Python package \texttt{ikpls}.