The Topological Stability Index: A Variance-Based Measure for Persistence Barcodes

We introduce the \emph{Topological Stability Index} (TSI), a variance-based scalar measure for persistence barcodes that quantifies the dispersion of persistence lifetimes. Unlike persistent entropy, which depends only on normalized weights, the TSI captures absolute variability and is sensitive to heterogeneous feature scales. We establish fundamental properties of the TSI, including its scaling behavior, invariance under lifetime translation and explicit update formulas under insertion and deletion of bars. We also consider a complementary first-moment-type quantity, the Topological Signal Index (TSigI), which captures the typical scale of persistence lifetimes and provides additional interpretability alongside the TSI. We further introduce a normalized version, $cv\text{TSI}$, which is scale invariant and admits an explicit algebraic relation to the Rényi entropy of order two. In particular, $cv\text{TSI}$ is an affine function of the collision probability $\sum_i p_i^2$, and therefore a monotone reparametrization of the Rényi entropy, providing a direct link between variance-based and entropy-based summaries in topological data analysis. Numerical experiments on synthetic data and stochastic time series demonstrate that the TSI captures structural variability complementary to entropy: it is relatively insensitive to deterministic trends, while responding strongly to stochastic fluctuations and variations in persistence magnitude.

Controlled Face Manipulation and Synthesis for Data Augmentation

Deep learning vision models excel with abundant supervision, but many applications face label scarcity and class imbalance. Controllable image editing can augment scarce labeled data, yet edits often introduce artifacts and entangle non-target attributes. We study this in facial expression analysis, targeting Action Unit (AU) manipulation where annotation is costly and AU co-activation drives entanglement. We present a facial manipulation method that operates in the semantic latent space of a pre-trained face generator (Diffusion Autoencoder). Using lightweight linear models, we reduce entanglement of semantic features via (i) dependency-aware conditioning that accounts for AU co-activation, and (ii) orthogonal projection that removes nuisance attribute directions (e.g., glasses), together with an expression neutralization step to enable absolute AU edit. We use these edits to balance AU occurrence by editing labeled faces and to diversify identities/demographics via controlled synthesis. Augmenting AU detector training with the generated data improves accuracy and yields more disentangled predictions with fewer co-activation shortcuts, outperforming alternative data-efficient training strategies and suggesting improvements similar to what would require substantially more labeled data in our learning-curve analysis. Compared to prior methods, our edits are stronger, produce fewer artifacts, and preserve identity better.