The problem begins with a formulation that has served statistics well for more than four decades. Cumulative-link ordinal models, introduced by McCullagh in 1980, classify ordered categories by placing a latent score on an axis and slicing that axis with learned thresholds. The probability of landing in any interior category is computed as the difference of two sigmoid functions evaluated at adjacent thresholds. This elegant construction underpins the proportional odds model, the CORAL ordinal regression framework, and softplus-reparameterized variants used throughout deep learning, from age estimation to histopathological grading. But the researchers point out an elementary consequence that is rarely confronted in the deep learning era: the maximum probability achievable for any interior stage is bounded by the gap between its two neighboring thresholds. The bound is tight and has a closed form, Pmax(g) = 2σ(g/2) − 1, and it applies to every model built on this sigmoid-difference formulation.

The numbers make the danger vivid. When the threshold gap equals 1.0, an interior category can never exceed a probability of 0.245, meaning the argmax decision rule will almost never select it, no matter how powerful the network, how long the training, or how abundant the data. At a gap of 2.0 the ceiling rises to 0.462, and at roughly 3.0 it reaches 0.635. The ceiling is a property of the sigmoid parameterization itself, and the authors note that it was already known in the psychometrics literature on graded response models, where Samejima studied the same category response function in 1969. What classical item-response theory never addressed, however, is what happens when such parameters are estimated by stochastic gradient descent on severely imbalanced image batches rather than by maximum likelihood on balanced matrices.

That imbalance is exactly where the failure mode becomes acute. In clinical datasets of peripheral blood cells, mature segmented neutrophils outnumber early-stage myelocytes by ratios exceeding 100 to 1. The dominant endpoint class sits far above all interior thresholds on the latent axis, so it contributes essentially no gradient at the interior boundaries. Meanwhile, the rare interior samples that would push those thresholds apart are too few, and too poorly positioned early in training, to supply the restoring force. The result is a self-reinforcing feedback loop: narrow gaps keep interior stages hard to place, and misplaced interior samples keep the gaps narrow. The researchers derive the threshold gradients explicitly and show that a correctly positioned interior sample exerts a widening force that grows monotonically as the gap shrinks and diverges as the gap approaches zero, a restoring mechanism that can rescue a collapsing configuration once features become separable.

To test these predictions, the team built HemoCline, a convolutional network of just 532,131 parameters whose backbone is assembled from mobile inverted bottleneck blocks, the same efficient building blocks behind the MobileNet family. On top of this shared backbone sits a hierarchical classification head that routes cells through biologically motivated branches, mirroring the taxonomy of granulocytes, agranulocytes, red cell lineage, and platelets, and then classifies granulocyte maturation stages through a cumulative-link head with learnable, strictly ordered boundaries enforced by softplus reparameterization. The four modeled stages follow the well-characterized neutrophilic maturation sequence from myelocyte to metamyelocyte to band neutrophil to segmented neutrophil, with each transition marked by progressive nuclear condensation and lobulation that human annotators discretize into stages.

The experimental validation ran on two independent public datasets: the KU-Optofil dataset from Turkey, containing 31,489 images across 13 classes with a striking 167 to 1 imbalance between its largest and smallest maturation classes, and the Barcelona PBC dataset from Spain with 17,092 images across 8 classes. Trained entirely from scratch without ImageNet pretraining, HemoCline reached 98.86 percent macro-F1 on Barcelona and 93.36 percent accuracy on KU-Optofil’s harder 13-class task. Under an identical from-scratch recipe, it matched MobileNetV3-Small on accuracy with 2.9 times fewer parameters while exceeding it on macro-F1, maturation-chain F1, calibration, and seed-to-seed stability. Against the best-performing pretrained model of the original KU-Optofil benchmark, DenseNet-121 at 13 times the parameter count, HemoCline sits 1.9 percentage points lower on accuracy, a gap the authors attribute honestly to capacity rather than claiming parity.

Perhaps the most scientifically interesting results are the ones the authors report against their own expectations. The study pre-registered four hypotheses, and two were falsified. The Maturation Coordinate, the scalar output of the ordinal head, was predicted to form a smooth continuum correlating strongly with true stage; instead it converged to four separated clusters, with a Spearman correlation of just 0.30 against a pre-specified threshold of 0.75. Yet the clusters discriminate powerfully between adjacent stages, with Cliff’s delta reaching nearly perfect separation for the metamyelocyte-to-band transition. The model, in other words, learns to distinguish stages without interpolating between them. The second falsification concerned cross-laboratory transfer: adjusting only the learned thresholds on data from a new laboratory recovered 76.5 percent macro-F1, far short of the 91 percent achieved by retraining the full classification head, because branch and sub-head probabilities shift between laboratories just as the maturation boundary does.

The threshold dynamics themselves behaved as the gradient analysis predicted, but with a twist. Starting from a deliberately narrow initialization with a gap of 1.0, where the ceiling analysis predicts near-zero interior recall, the optimizer self-widened the gaps to an average of about 2.0 within 40 epochs, confirming the restoring-force mechanism. But the gap-floor regularizer the team derived from the closed-form minimum gap formula turned out to be inactive at the working initialization, where gaps already sat above the floor, and it failed to rescue a shortened training schedule. An ablation against CORN, an ordinal method that avoids the ceiling by construction, was statistically indistinguishable from the cumulative-link head across five seeds. The authors conclude that the ceiling is real and observable at narrow initializations, but that the correct response is a wider initialization and monitoring of gap trajectories rather than an added loss term, and they offer the closed-form minimum gap as a diagnostic grounded in the stage count rather than a grid search.

The study also distinguishes carefully between the two head configurations and what each is for. On aggregate metrics the flat, non-ordinal head matches or slightly beats the ordinal one, and it wins on agranulocyte separation, distinguishing lymphocytes from reactive lymphocytes and blasts more reliably. The ordinal head earns its place on the sub-task it was designed around: maturation-stage resolution improves, with quadratic weighted kappa rising from 0.815 to 0.877, and the hardest class, band neutrophil, shows three times lower seed-to-seed variability. The choice between them, the authors argue, is a choice about which errors a deployment can absorb. All configurations passed the team’s multi-seed deployment-readiness protocol, which trains five copies differing only in random seed and requires both low coefficient of variation and no seed below pre-committed performance floors.

The practical implications reach beyond hematology. The interior-stage bound applies to any model computing category probabilities as differences of sigmoid threshold terms, and the authors suggest that tasks with more ordinal stages, where the same latent axis must accommodate more boundaries and tighter gaps, are where the collapse phenomenon should matter most. For point-of-care devices operating under tight memory and latency budgets, a 532 K-parameter network that runs in under 2 milliseconds on CPU offers a credible alternative to the massive pretrained backbones that dominate the literature. The authors are candid about limitations: myelocyte classification remains poor for both configurations with only 68 training examples, the transfer analysis spans just two laboratories, and clinical deployment would require prospective validation and regulatory clearance. But as a demonstration that a small, biologically structured model can approach the accuracy of giants while exposing a hidden mathematical trap in the process, the work marks a compelling step toward AI that understands not just what a cell is, but where it sits along the road of its own maturation.

Subject of Research: Threshold gap dynamics in cumulative-link ordinal models for imbalanced blood cell maturation classification

Article Title: HemoCline: Threshold gap dynamics in cumulative-link ordinal models for imbalanced blood cell maturation

Article References: Daga, M., Bommineni, K. R., & Ramu, S. P. (2026). HemoCline: Threshold gap dynamics in cumulative-link ordinal models for imbalanced blood cell maturation. Machine Learning with Applications, 26, Article 101031. https://doi.org/10.1016/j.mlwa.2026.101031

Image Credits: AI Generated

DOI: 10.1016/j.mlwa.2026.101031

Keywords: ordinal regression, cumulative-link models, blood cell classification, class imbalance, neutrophil maturation, HemoCline, lightweight neural networks, hematology, deep learning, threshold dynamics, medical imaging, machine learning