LoomSC: Scalable Deep Subspace Clustering with Projector Factorization and Exact Spectral Reduction

We derive the linear union-of-subspaces (UoS) model for subspace clustering (SC) from the nonlinear mixture model (NMM) used in blind source separation (BSS) to represent a D-dimensional observation vector as an unknown multivariate nonlinear mapping of C latent variables. Assuming the mapping is differentiable up to an unknown order K, we approximate NMM by a K-th order Taylor expansion, yielding a model equivalent to the linear UoS framework underlying SC. This establishes that: (i) the smoothness order K corresponds to the unknown subspace dimension d; (ii) KC equals the number of anchors; and (iii) the sparsity of the representation vector equals K (i.e., d). These relationships enable estimation of bounds on subspace dimension, and that is validated on six benchmark datasets using five established SC algorithms. Established theoretical results are important for post-processing of self-representation matrices estimated by SC algorithms.

Accurate optimization of a supervised spectral objective need not produce an accurate population subspace or a better predictive representation. We investigate these distinctions for Online Kernel Supervised Principal Component Analysis (OKSPCA), which combines a centered cross-moment in finite random-feature coordinates with an Adam-style orthonormal basis update for an established objective. Fixed-map consistency, concentration and perturbation results describe the estimator and its exact subspace; same-target comparisons then assess the practical iterate separately. Across six predictive benchmarks, performance depends on the declared pipeline: replacing the tracker with the exact empirical target leaves the two regression deficits largely unchanged. Direct classification-rank models capture nearly all terminal objective energy on average, but a saved intermediate state exhibits substantial geometric deviation; a controlled sample-size study further separates empirical accuracy from population recovery. In distinct numerical-service workloads, exact on-request computation is faster in the tested classification settings, whereas Adam saves time relative to the tested full thin-SVD service for some dense wider-regression requests, alongside persistent geometric error. These diagnostics limit explanations based solely on terminal optimization accuracy and distinguish numerical cost from quality, rank coverage and freshness; they establish neither practical-tracker convergence nor predictive or deployment benefits from basis availability.

Scaling LLM-based applications to millions of users is bottlenecked by the inference cost and latency of modern foundation models. A natural fix is to cluster the inputs and call the LLM only on cluster representatives, letting other members inherit the output -- but this is only safe if each member is measurably close to its representative. Existing clustering methods do not offer such per-sample quality control at scale: none jointly guarantee a minimal within-cluster similarity, exact matching of categorical attributes, and scalability to tens of millions of samples. We propose a two-stage algorithm that generates initial clusters with Mini-batch K-Means, then greedily selects representatives within each initial cluster -- a step equivalent to the Johnson-Chvatal heuristic for Set Cover over alpha-balls in embedding space. The algorithm enforces the similarity and attribute guardrails exactly by construction, and runs in $O(nd + n^2 d/K)$ time and $O(nd + n^2/K^2)$ memory for $n$ samples, feature dimension $d$, and $K$ initial clusters -- linear in $n$ when $K$ grows proportionally with $n$. We provide benchmarks against common clustering methods on internal and public datasets: our method not only delivers per-sample guardrails but also runs 10-1000x faster and scales to data sizes where most standard methods become intractable. Deployed on 38 million customers for a persona-based recommender, the clustering method cut downstream cost and latency by 50-fold while preserving personalization and unblocked the production launch.