Links:

• Preprint: https://doi.org/10.5281/zenodo.22028087
• Entropic Scree Function v1.0.0 / GitHub: https://github.com/tjleestjohn/Entropic-Scree

TL;DR:

Standard PCA fundamentally fractures non-linear dependencies into "Spurious Orthogonal Dimensions," drastically overestimating the true rank of complex tabular systems. Meanwhile, non-linear alternatives like Kernel PCA and Euclidean nearest-neighbor estimators suffer structural collapse when generative roots are entangled or sparse.

I’m sharing the methodology and code here for anyone dealing with these complex tabular data nightmares.

The method and open-source framework use Normalized Mutual Information to compress spurious expansions back towards their true generative roots. It also

• Maps the underlying "informational gravity" of the roots, offering insight into overall average stability, as well as which specific roots can be most reliably extracted;
• Estimates the data's overall ratio of shared signal to unshared idiosyncratic informational variance (noise);
• Serves as a powerful exploratory map that separates unrelated clusters of variables, allowing you to easily identify decoupled sub-networks.

A Modern ML Architectural Blueprint: Far beyond a mere update to legacy factor analysis workflows, identifying this exact intrinsic rank allows you to explicitly size neural bottlenecks for downstream non-parametric manifold extractors (like autoencoders).

The Problem with Standard Baselines:

When trying to map the intrinsic dimensionality of a dataset, standard practice usually dictates reaching for PCA, its non-linear kernel extensions, or Euclidean nearest-neighbor estimators. But if your tabular environment has mixed data types, heavy non-linearities, entangled roots, or more features than samples ($m > N$), these established baselines don't just lose precision. They suffer a structural collapse.

The core issue with our standard baselines:

• Standard PCA drives Dimensional Inflation. Because it only measures linear covariance, it perceives a polynomial expansion or a non-linear interaction (like $X_1 X_2$) as an entirely independent variable. It is forced to fabricate new, spurious orthogonal dimensions to map them.
• Kernel PCA (RBF) suffers Structural Collapse. Projecting into a Hilbert space doesn't fix this. KPCA artificially folds even-polynomials into independent axes. Furthermore, because its infinite-dimensional space lacks a finite-sample boundary, sparse combinatorial noise smears into an elevated tail that obscures the structural elbow. If the underlying generative roots are even mildly entangled, KPCA suffers a total structural collapse.
• Topological Estimators (Euclidean) fail in sparse regimes. Estimators like TWO-NN or MLE rely on Euclidean distance metrics. In asymmetric, feature-rich environments ($m > N$), they suffer from distance concentration (the ratio between nearest and farthest neighbors converges to 1). This renders local neighborhood calculations structurally degenerate across mixed-data margins.

Introducing the Entropic Scree:

To solve this, I built the Entropic Scree. It throws out linear and spatial variance entirely and evaluates pure probability mass.

Here is how it works under the hood:

1. The Metric Space: It evaluates pairwise dependencies using Information-Theoretic Jaccard Similarity (Variation of Information). Because this relies on Shannon entropy, it’s invariant to marginal shape mismatches (like mixing continuous waves with binary flags).
2. Bypassing the Rank Ceiling: Standard PCA is algebraically capped at $N-1$. By moving to a double-centered topological information space, we map true overlapping redundancy and completely bypass the algebraic sample-size ceiling.
3. Compressing the Manifold: The algorithm acts as a bivariate filter. It inherently compresses the primary overlapping probability mass of non-linear combinations back towards the Intrinsic Generative Rank. It shears off the unique synergistic variance, leaving behind residuals that form a bounded Extended Signal Tail, cleanly separating the true drivers from the unstructured Idiosyncratic Informational Variance.

Quantifying Informational Gravity:

Beyond just extracting a discrete rank, the framework decouples rank from probabilistic volume by introducing Informational Gravity (AIG/FSIG). By systematically rebundling the residual variance sheared off by the bivariate filter, it translates abstract matrix properties into actionable, "variable-equivalent" footprints.

Empirical Stress Test:

To demonstrate the theoretical bounds, I built a highly entangled synthetic dataset with 20 pure generative roots expanded into 5th-order combinatorics across 20,000 proxies, but only 10,000 samples ($m > N$). To truly simulate messy, real-world contexts, I also heavily injected idiosyncratic structural noise and measurement error into the data.

• Standard PCA hit the rank ceiling, linearly fractured the expansions, and falsely extracted ~5,700 dimensions.
• Kernel PCA (RBF) & Spearman Rank structurally folded and yielded a liberal overestimation of the rank by 100%. When root entanglement was introduced, they completely lost their elbows and suffered total structural collapse.
• The Entropic Scree correctly mapped the intrinsic rank at exactly 20. It successfully isolated a mere 1.45% of active shared signal from an overwhelming 98.55% bulk of unstructured Idiosyncratic Informational Variance. Furthermore, the residuals formed an Extended Signal Tail that perfectly aligned with the deterministic limits of the global hypergeometric design space.
• Mapping Hidden Topology: Using Factor-Specific Informational Gravity (FSIG), the framework successfully reverse-engineered the simulation's hidden architecture. The topology profile diagnosed a large primary dimension ($FSIG_1 \approx 74.5$ variable equivalents) mapping the network's global combinatorial hub, followed immediately by a flat plateau across the remaining 19 dimensions ($\sim 11.5$ each), confirming a democratically distributed root system beneath the extreme entanglement.

Feedback / Discussion:

How are you currently handling intrinsic rank extraction in these messy, complex tabular environments?

If you are wrestling with sample-starved, heavily non-linear generative datasets where standard PCA and other baseline tools just aren't cutting it, I’d love for you to pull the Entropic Scree repo and test it yourself.

I'm completely open to feedback, so let me know how it performs for you and I'm happy to discuss the mechanics.