From Tsallis to KL: Convergence and Error Estimates for Tsallis-Regularized Optimal Transport

Scientific Impact Assessment

Core Contribution

This paper provides a rigorous analytical treatment of the Tsallis-to-KL limit for entropy-regularized optimal transport (OT). Three contributions stand out. First, the authors derive an *exact variational reformulation* (Proposition 1.1) expressing Tsallis-regularized OT in terms of a Tsallis information projection onto the coupling set, isolating an explicit correction term. This cleanly explains *why* — unlike the special KL case — the regularized transport problem and the information projection problem do not coincide for general Tsallis divergences. Second, they establish existence and uniqueness for the Tsallis information projection. Third, they prove Γ-convergence (in the narrow topology) of Tsallis-regularized functionals to the KL functional as q↓1, with narrow convergence of minimizers, and — most valuably — explicit O(q−1) error estimates for both regularized OT values and information projection values, with constants depending explicitly on ‖c‖∞ and ε.

The work fits a niche: it complements the authors' prior study of the ε↓0 limit [22] by instead fixing ε and taking q↓1. The correction-term formula is the conceptual centerpiece, giving structural insight into the divergence between f-divergence regularization and f-projection.

Methodological Rigor

The mathematical development is sound and self-contained. Proofs are careful: the Γ-convergence argument correctly handles liminf/limsup inequalities with an explicit recovery-sequence construction (truncation + marginal correction) and a diagonal-sequence argument to combine n→∞ and q_k↓1. Lower semicontinuity, narrow compactness of the coupling set, and strict convexity of the generator ϕ_q are appropriately invoked for existence/uniqueness. The error estimates rely on clean pointwise inequalities (mean value theorem bounds on t^q−t, log₂₋q(x)−log x) and uniform L∞ bounds on optimal couplings via Schrödinger-potential bounds imported from Di Marino–Gerolin [11]. The constructions are convincing, and the constants are explicit rather than abstract, which is a strength. The results are restricted to 1<q≤2 and nonnegative bounded continuous costs — reasonable and clearly stated assumptions. No gaps are apparent in the derivations.

Potential Impact

The impact is likely to be modest and confined to the theoretical OT / information-geometry community. Entropy-regularized OT is a heavily used tool in ML, imaging, and graphics, and generalized (Tsallis / f-divergence) regularizations have appeared in the ML literature (Muzellec et al. AAAI 2017, Bao–Sakaue, Terjék–González-Sánchez AISTATS 2022). Quantifying the Tsallis-to-KL passage with explicit rates provides a principled justification for using Tsallis regularization as a tunable interpolation and for understanding numerical behavior as q→1. However, the paper contains no algorithms, no experiments, and no direct computational contribution. Its influence will be primarily as a cited theoretical reference for those working on generalized regularizers, sparsity via q-entropy, or the projection-vs-regularization distinction. The O(q−1) rate is a useful concrete takeaway but is not surprising — first-order convergence in the deformation parameter is the expected outcome.

Timeliness & Relevance

Generalized-entropy OT is an active but specialized topic. The paper addresses a genuine conceptual question (why regularized OT and information projection diverge for non-KL divergences) that had been noted but not sharply quantified. It is timely within its subfield but does not address a widely-felt bottleneck; the machine-learning demand for Tsallis OT specifically is limited compared to KL/Sinkhorn.

Strengths & Limitations

Strengths:

Limitations:

Other Observations

This is a "complete-in-itself" analytical result rather than a reusable building block or framework. It corroborates the intuitive expectation that Tsallis→KL smoothly, adding quantitative precision. Reproducibility in the theoretical sense is high — all proofs are given in full. There is no dataset, code, or empirical component. The paper does not contest any prior claim; it clarifies and quantifies a known structural distinction. Its foundational value is limited: others may cite the correction-term formula and rate, but few will build machinery directly on top of it.

Overall, a technically clean, well-executed piece of specialized mathematical analysis with clear but bounded significance.