Sketch-orthogonal low-rank optimizer updates

Integrate this at the low-rank weight or optimizer-state update, not at the dense matrix multiplication itself. Represent a selected linear-layer weight as $W=W_0+UV^\top$, with $U\in\mathbb{R}^{d_{out}\times r}$ and $V\in\mathbb{R}^{d_{in}\times r}$, and obtain the dense gradient $G=\partial L/\partial W$ either explicitly for a small benchmark or through matrix-free products. At each update, first form the tangent-projected gradient $H=P_{U,V}(G)$ using the displayed projector, then update $U,V$ with the chosen factor-gradient rule or with a rank-$r$ factorization of $H$. Every $K$ steps, draw a fixed or refreshed sketch $S\in\mathbb{R}^{s\times d_{out}}$ with $s=r+8$ or $2r$, compute $Y=SU$, $C=Y^\top Y+\epsilon I$, and the Cholesky factor $R=\operatorname{chol}(C)$. Replace $U$ by $UR^{-1}$ and $V$ by $VR^\top$; this preserves the represented low-rank matrix while reconditioning the basis. In distributed training, compute $SU$ using an all-reduce on only the $s\times r$ sketch Gram matrix rather than a full QR synchronization. Measure $\|U^\top U-I\|_F$, projector residual $\|G-H\|_F$, communication volume, wall-clock time, and loss. First test rank-8 and rank-16 LoRA on one Transformer MLP or attention projection using RoBERTa-base on GLUE, comparing ordinary QR, no reorthogonalization, and sketch-Cholesky QR. A successful signal is equal validation accuracy and loss trajectory with lower orthogonalization time or communication, while the no-reorthogonalization baseline develops worse conditioning.