The problem the paper addresses is deceptively simple. Statisticians routinely fit regression models with an intercept, allowing the line to pass through the centroid of the data cloud. But many scientific questions demand a no-intercept, or homogeneous, model, in which the response must vanish when the predictor vanishes. Metabolic rate, for instance, should arguably fall to zero as body mass falls to zero. The default strategy for switching between these specifications has always been to refit the model from scratch, which implicitly assumes the analyst can return to the full design matrix and response vector. In legacy studies, privacy-restricted medical research, and federated learning systems, that assumption often fails.

Selvitella’s central theorem shows that for any quadratic-loss regression, including ordinary least squares, ridge regression, and weighted least squares, the no-intercept estimator can be written in closed form as a function of the with-intercept coefficients, the empirical means of the variables, and the Gram matrices that summarize the data’s geometry. The key identity decomposes the uncentered cross-products used by the no-intercept model into centered cross-products plus a mean-product term. Because the with-intercept solution already satisfies a centered normal equation, the no-intercept solution can be recovered by a single matrix inversion that depends only on the design, the weighting, and any penalty structure, never on the response itself.

Just as importantly, uncertainty propagates through the same bridge. The no-intercept estimator turns out to be an affine function of the with-intercept estimator, so standard covariance calculus delivers its variance directly. Under Gaussian errors, the transformed estimator exactly reproduces the finite-sample distribution of the classical no-intercept solution, and under weak moment conditions it inherits the usual asymptotic normality through Slutsky’s theorem. In other words, confidence intervals and hypothesis tests for the through-the-origin model can be reconstructed from published summaries alone, an algebraically exact result rather than an approximation.

The first major application is federated learning, where data are distributed across many clients, such as hospitals or mobile devices, that never share raw observations. If a consortium has already trained a with-intercept model through the popular FedAvg protocol and then decides it needs the no-intercept version, the conventional answer is a second full federated run, costing on the order of the number of rounds times the sample size times the predictor dimension in local computation, plus repeated communication rounds. The new theorem shows the server can instead aggregate one round of local sufficient statistics, each client transmitting only its sample count, moment vectors, and Gram matrix, and then compute the no-intercept estimator with a cost that scales with the predictor dimension alone and is independent of the number of clients, observations, or communication rounds.

Numerical experiments in the paper back the theory with striking precision. In simulations with ten clients and five thousand observations, the transformed estimator agreed with the centralized no-intercept ordinary least squares solution at floating-point precision, around ten to the minus fifteen. The direct federated no-intercept baseline, by contrast, drifted from the centralized answer as client data became heterogeneous, with relative errors growing from roughly four ten-thousandths in the balanced case to a full ten percent under severe non-IID conditions, a manifestation of the well-known client-drift problem. Timing measurements showed the transformation running between one thousand and twenty-three thousand times faster than a second federated training run, and the agreement held at machine precision even in high-dimensional settings with up to five hundred predictors.

The paper is careful about scope. The savings are relative to a second iterative federated optimization, not to one-shot protocols in which clients transmit exact Gram matrices from the outset, and the method is not by itself a formal privacy guarantee. Aggregate summaries can still leak information when a client holds few observations or unusual covariate patterns, so the author notes that secure aggregation or calibrated noise would need to be layered on top where formal differential privacy is required. There is also a communication trade-off: the one-shot summary protocol costs on the order of the predictor dimension squared per client, so it beats a second federated run only when the number of additional rounds exceeds roughly a quarter of the predictor dimension, a break-even the paper quantifies explicitly.

The second implication is more conceptually surprising: a regression reversal phenomenon. In simple linear regression, the slope estimated with an intercept and the slope estimated through the origin can have strictly opposite signs. Selvitella proves an exact necessary and sufficient condition, namely that the centered cross-product and the uncentered cross-product have opposite signs, which happens precisely when the mean-product term dominates the centered signal. The reversal is driven by the intercept constraint and by leverage, not by classical confounding, yet it is structurally analogous to Simpson’s paradox: in both cases an unadjusted association can point opposite to an adjusted one, with the omitted constant term playing the role of a lurking group-level offset. Because the condition can be checked from summary statistics alone, practitioners can detect potential sign flips before committing to a refit.

The third implication reaches into biology, where only aggregate results are often available and the intercept carries real scientific weight. Reanalyzing a published study of basal metabolic rate in sixty-eight adult men, the paper reconstructs the through-the-origin fit entirely from reported means, standard deviations, and the correlation coefficient. The no-intercept slope comes out near 0.0975, nearly double the with-intercept slope of about 0.054, illustrating how a nonzero intercept can attenuate an estimated scaling relationship. Goodness-of-fit measures can also be reconstructed, though the paper cautions that the uncentered coefficient of determination for no-intercept models is not directly comparable to the conventional centered one; on a common centered scale, the no-intercept model explains less variance than the with-intercept fit, consistent with the original authors’ conclusion that simple linear models inadequately capture allometric scaling.

The broader message is that a piece of linear algebra most statisticians thought they already understood still holds surprises with practical teeth. By placing the affine-to-linear map within a unified framework covering penalized and weighted losses, and by propagating both point estimates and their distributions, the result turns intercept-specification changes from an expensive refitting problem into a cheap post-processing step. For meta-analysts working with decades of published summaries, for consortia bound by data-use agreements, and for engineers designing federated systems, the ability to move between model specifications using sufficient statistics alone offers a rare combination of mathematical exactness and immediate utility, and it suggests that other transformations across the regression landscape may be waiting for similar treatment.

Subject of Research: Summary-statistics-only affine-to-linear transformations linking with-intercept and no-intercept quadratic-loss regression, with applications to federated learning, regression reversal, and biological scaling

Article Title: Affine-to-linear transformations for quadratic-loss regression: computational implications to federated learning, reversal paradox, and applications

Article References: Selvitella, A. M. (2026). Affine-to-linear transformations for quadratic-loss regression: computational implications to federated learning, reversal paradox, and applications. International Journal of Data Science and Analytics, 22(1), Article 289. https://doi.org/10.1007/s41060-026-01268-6

Image Credits: AI Generated

DOI: 10.1007/s41060-026-01268-6

Keywords: regression, federated learning, sufficient statistics, ordinary least squares, ridge regression, weighted least squares, intercept, Simpson's paradox, metabolic scaling, covariance propagation, communication cost, data privacy

The core insight behind EPR-C3 is that the search for the best subset should not be blind. Traditional approaches to the problem fall into two broad camps, each with well-known weaknesses. Stepwise procedures, which add or remove predictors one at a time based on statistical criteria, are fast but notoriously unstable, often missing the globally best model and producing results that shift with small perturbations of the data. Penalized regression methods such as the lasso, ridge, and elastic net shrink coefficients to perform implicit selection, but they transform the problem itself, replacing the ordinary least squares formulation with a modified objective and yielding biased estimates that complicate interpretation. Genetic algorithms and other stochastic search techniques can explore large spaces, but their reliance on random initialization and random operators means that two runs of the same algorithm on the same data can produce different answers, undermining the reproducibility that scientific work demands.

EPR-C3 takes a different path. The method is a deterministic multi-start neighborhood-search heuristic, meaning that it systematically explores multiple regions of the subset space using structured moves rather than random jumps, and that repeated runs on identical data produce identical results. The name encodes its workflow: expansion, perturbation, and reduction form the exploratory core, while C3 refers to a three-part constraint refinement stage comprising correlation cleanup, replacement recovery, and variance inflation factor pruning. Each stage addresses a specific pathology of high-dimensional regression. Expansion grows a candidate subset by adding predictors that promise improvement. Perturbation shakes up the current solution to escape local optima, the traps where a greedy search would otherwise stall. Reduction trims away predictors that no longer earn their place.

The C3 refinement stage is where the method embeds statistical admissibility directly into the search. Correlation cleanup removes predictors that are so strongly correlated with others already in the model that they add redundancy rather than information. Replacement recovery attempts to swap problematic predictors for alternatives that preserve predictive power while restoring stability. Variance inflation factor pruning, the final safeguard, quantifies how much the variance of an estimated coefficient is inflated by multicollinearity and eliminates predictors whose presence destabilizes the model. By enforcing these constraints during the search rather than checking them afterward, EPR-C3 ensures that every candidate model it evaluates is statistically admissible, avoiding the embarrassing situation in which a search returns a model with excellent fit but coefficients so unstable that they are meaningless.

A crucial design decision sets EPR-C3 apart from penalized approaches: it preserves the native ordinary least squares formulation throughout. The final output is not a shrunken or regularized set of coefficients but an explicit regression equation of the familiar form, with interpretable coefficients estimated by classical least squares. This matters enormously for applied researchers in fields such as chemistry, medicine, and the social sciences, where the goal is often not just prediction but understanding, and where an explicit equation with transparent coefficients is the deliverable that matters. The author’s own background in chemical modeling, including prior work on predicting acid dissociation constants in nitrogen compounds, reflects this applied orientation toward interpretable equations.

The benchmark results are striking. In analyses comparing EPR-C3 against exhaustive all-subsets regression, the heuristic recovered a high proportion of the Top-K admissible subsets, the best-ranked models that exhaustive search identifies, while dramatically reducing computational cost in highly combinatorial problems. In other words, when the exhaustive answer is computable, EPR-C3 usually finds it; when exhaustive search is impossible, EPR-C3 still delivers high-quality admissible models in a fraction of the time. The study goes further and proposes an empirical utility threshold, a practical guideline that tells analysts when the heuristic search becomes preferable to exhaustive enumeration, giving practitioners a principled rule for choosing between the two strategies rather than guessing.

The comparative evaluation is unusually thorough for a methods paper. EPR-C3 was benchmarked against stepwise procedures, penalized-regression preselection, filter and wrapper screening approaches, genetic algorithms, and branch-and-bound subset selection, the latter being the classical exact method that prunes the search space using bounds on the objective. Across these comparisons, a consistent pattern emerged: embedding admissibility constraints within the search workflow improves the tradeoff between recovery of the best models and computational efficiency. Competing methods that search first and check statistical validity later, or that abandon the least squares framework entirely, tend to either waste effort on inadmissible candidates or return models that require post-hoc repair. EPR-C3’s constraint-aware architecture avoids both failure modes.

Reproducibility and robustness received dedicated attention. Bootstrap analyses, in which the method was run on resampled versions of the data, showed close agreement with exhaustive selection in best-subset recovery, out-of-bag performance, and predictor-inclusion concentration, the degree to which the same predictors are selected consistently across resamples. This concentration measure is particularly important because instability of variable selection is one of the most cited criticisms of heuristic and stepwise methods. A seed-package analysis further confirmed that the deterministic nature of the algorithm holds in practice, with repeated runs converging on the same solutions. An applied case study demonstrated that EPR-C3 could reproduce a previously published multiple linear regression equation, recovering the same model with reduced runtime, providing a concrete demonstration that the method delivers on real problems rather than only on synthetic benchmarks.

The significance of this work extends beyond the specific algorithm. Variable selection remains one of the most contested practices in applied statistics, with methodologists repeatedly warning that automated selection can produce misleading inference, inflated significance, and models that fail to replicate. By making the search deterministic, constraint-aware, and anchored in the classical least squares framework, EPR-C3 offers a middle path between the computational impossibility of exhaustive search and the statistical compromises of penalization and greedy stepwise selection. The method is implemented in the author-maintained MLR-X project and publicly available, lowering the barrier for adoption. As datasets in genomics, chemometrics, economics, and the behavioral sciences continue to swell in dimensionality, tools that combine computational tractability with statistical admissibility and exact reproducibility are likely to find eager audiences. The study, published as volume 22, article 311 of the journal and supported by ANID-FONDECYT grant funding, suggests that the future of high-dimensional model selection may lie not in abandoning classical regression but in searching its vast model space more intelligently.

Subject of Research: Deterministic constraint-aware subset selection in high-dimensional multiple linear regression

Article Title: Epr-c3: a deterministic constraint-aware heuristic for high-dimensional subset selection in multiple linear regression

Article References: Epr-c3: a deterministic constraint-aware heuristic for high-dimensional subset selection in multiple linear regression. (n.d.). https://doi.org/10.1007/s41060-026-01298-0

Image Credits: AI Generated

DOI: 10.1007/s41060-026-01298-0

Keywords: subset selection, multiple linear regression, ordinary least squares, multicollinearity, variance inflation factor, heuristic optimization, best-subset regression, statistical computing, high-dimensional data, model selection, deterministic algorithms, bootstrap validation