KAPLAN-HR: Nonparametric Hazard Regression

• KAPLAN-HR is a survival-analysis model that leverages Kolmogorov-Arnold Networks with B-spline activations to model the conditional hazard for right-censored data.
• It bridges classical spline-based generalized additive models and deep survival models by capturing non-additive, time-varying interactions without manual feature engineering.
• Theoretical analysis shows convergence rates based on function smoothness, mitigating the curse of dimensionality and enhancing predictive performance in high-dimensional settings.

Searching arXiv for KAPLAN-HR and closely related survival-analysis KAN papers. I’m looking up recent arXiv entries on KAPLAN-HR, KANs for survival analysis, and closely related Kaplan–Meier / hazard-estimation work to ground the article in current literature. KAPLAN-HR is a survival-analysis model introduced in "KAPLAN: Kolmogorov-Arnold Prognostic Learnable Activation Networks for Survival Analysis" (Logothetis et al., 21 May 2026). It uses a Kolmogorov-Arnold Network with B-spline-parameterized activations to estimate the conditional hazard as a joint, nonparametric function of covariates and time under right censoring. The model is positioned between classical spline-based generalized additive models and deep neural survival models: a single-layer KAPLAN-HR model recovers a generalized additive proportional hazards specification, whereas deeper architectures capture interactions and time-varying effects through composition. Its central theoretical claim is a convergence rate for the nonparametric KAN hazard estimator that depends on the smoothness of the underlying KAN representation rather than on covariate dimension, thereby mitigating the curse of dimensionality for KAN-representable targets (Logothetis et al., 21 May 2026).

1. Survival setting and modeling objective

KAPLAN-HR is formulated for right-censored data

$$\mathcal{D}=\{(\mathbf{x}_i,Y_i,\Delta_i)\}_{i=1}^n,$$

where $\mathbf{x}_i\in[0,1]^d$ are covariates, $Y_i=\min\{T_i,C_i\}\in[0,1]$ is the observed time after rescaling to a finite study horizon, and $\Delta_i=\mathbb{1}\{T_i\le C_i\}$ is the event indicator. The conditional hazard is written as

$$\lambda(t\mid \mathbf{x})=-\frac{d}{dt}\log S(t\mid \mathbf{x}),$$

with independent censoring given covariates, $T\perp C\mid X$ (Logothetis et al., 21 May 2026).

The methodological motivation is the inadequacy of manual specification in classical survival models when datasets become rich and high-dimensional. In the Cox model,

$$\lambda_{\text{PH}}(t\mid \mathbf{x})=\lambda_0(t)\exp(g(\mathbf{x};\theta)),$$

covariates enter only through a time-invariant log-partial hazard, so time-varying effects and interactions must be hand-engineered. Spline-based GAMs and HARE similarly require additive terms or explicitly specified tensor-product or pairwise interaction structures. KAPLAN-HR instead models the log-hazard as a joint function of covariates and time,

$$\lambda_{\text{HR}}(t\mid \mathbf{x})=\exp(g(\mathbf{x},t)),$$

so that interactions and non-proportional hazards can be represented directly by the architecture rather than by feature construction (Logothetis et al., 21 May 2026).

A central implication is that KAPLAN-HR is not merely a replacement for the Kaplan-Meier estimator or for Cox partial-likelihood estimation. It is a continuous-time hazard-regression model trained on the full right-censored likelihood, with time treated as an explicit input coordinate rather than as a nuisance axis or baseline component (Logothetis et al., 21 May 2026).

2. Hazard parameterization and KAN architecture

KAPLAN-HR adopts a hazard-regression parameterization

$$\lambda_{\text{HR}}(t\mid \mathbf{x})=\exp\!\left(g(\mathbf{x},t;\boldsymbol{\theta})\right),$$

where $g(\mathbf{x},t;\theta)$ is the log-hazard. The model instantiates $\mathbf{x}_i\in[0,1]^d$0 as a Kolmogorov-Arnold Network (KAN), so the input is the concatenated vector

$\mathbf{x}_i\in[0,1]^d$1

and the network output is scalar: $\mathbf{x}_i\in[0,1]^d$2 Each layer computes

$\mathbf{x}_i\in[0,1]^d$3

so nodes sum their inputs while edges apply trainable univariate functions $\mathbf{x}_i\in[0,1]^d$4 (Logothetis et al., 21 May 2026).

These edge functions are parameterized by B-spline expansions, augmented in implementation by a scaled SiLU residual. If $\mathbf{x}_i\in[0,1]^d$5 denotes a spline on interval $\mathbf{x}_i\in[0,1]^d$6, the paper uses the bounds

$\mathbf{x}_i\in[0,1]^d$7

and

$\mathbf{x}_i\in[0,1]^d$8

with $\mathbf{x}_i\in[0,1]^d$9 depending only on spline order and $Y_i=\min\{T_i,C_i\}\in[0,1]$0 the mesh size. These spline regularity bounds are later used in approximation and entropy arguments (Logothetis et al., 21 May 2026).

The first layer admits a useful factorization. For each hidden unit $Y_i=\min\{T_i,C_i\}\in[0,1]$1,

$Y_i=\min\{T_i,C_i\}\in[0,1]$2

This separates covariate and time contributions before they are recombined by deeper layers. Computationally, $Y_i=\min\{T_i,C_i\}\in[0,1]$3 and $Y_i=\min\{T_i,C_i\}\in[0,1]$4 can be evaluated separately and broadcast across a time grid, which reduces the cost of repeated hazard evaluations during numerical integration (Logothetis et al., 21 May 2026).

3. Relation to GAMs, proportional hazards, and interaction modeling

A defining structural property of KAPLAN-HR is that network depth controls the degree of departure from additive proportional hazards. When the architecture has a single layer, the log-hazard becomes

$Y_i=\min\{T_i,C_i\}\in[0,1]$5

so that

$Y_i=\min\{T_i,C_i\}\in[0,1]$6

This is structurally a proportional hazards model with a nonparametric baseline hazard and nonparametric additive spline effects in the covariates. The paper states explicitly that a single-layer KAPLAN-HR model recovers a GAM (Logothetis et al., 21 May 2026).

Deeper architectures remove that restriction. For $Y_i=\min\{T_i,C_i\}\in[0,1]$7, second-layer edge functions act on nonlinear combinations of time and multiple covariate components, for example

$Y_i=\min\{T_i,C_i\}\in[0,1]$8

This composition induces non-additive covariate interactions and time-covariate interactions without manual specification. In the terminology of the paper, deeper KAPLAN-HR architectures capture interactions and time-varying effects through composition (Logothetis et al., 21 May 2026).

This places KAPLAN-HR on a continuum. At one end, it coincides with spline-based additive PH modeling; at the other, it behaves as a flexible deep hazard model. A plausible implication is that architectural depth can be interpreted as a statistical complexity dial: shallow models preserve interpretability and PH structure, whereas deeper models permit non-proportional hazards and higher-order interactions.

4. Likelihood, numerical estimation, and asymptotic theory

The model is trained by minimizing the full right-censored negative log-likelihood

$Y_i=\min\{T_i,C_i\}\in[0,1]$9

where

$\Delta_i=\mathbb{1}\{T_i\le C_i\}$0

Because the cumulative hazard does not have a closed form for a generic KAN output, it is approximated numerically by right-endpoint Riemann sums on a time grid $\Delta_i=\mathbb{1}\{T_i\le C_i\}$1: $\Delta_i=\mathbb{1}\{T_i\le C_i\}$2 with corresponding discrete survival estimate

$\Delta_i=\mathbb{1}\{T_i\le C_i\}$3

Training uses Adam, mini-batches, early stopping on validation NLL, and an $\Delta_i=\mathbb{1}\{T_i\le C_i\}$4 shrink penalty on the log-hazard values evaluated over batch-specific time grids (Logothetis et al., 21 May 2026).

The theoretical analysis assumes that the true log-hazard $\Delta_i=\mathbb{1}\{T_i\le C_i\}$5 is bounded and belongs to a class of KAN-representable functions $\Delta_i=\mathbb{1}\{T_i\le C_i\}$6, where each edge function has Sobolev smoothness $\Delta_i=\mathbb{1}\{T_i\le C_i\}$7, network depth and widths are fixed, and the censoring law is well behaved on $\Delta_i=\mathbb{1}\{T_i\le C_i\}$8 with a point mass at $\Delta_i=\mathbb{1}\{T_i\le C_i\}$9 to accommodate administrative censoring. Estimation is performed over a sieve of spline-based KANs with knot count $$\lambda(t\mid \mathbf{x})=-\frac{d}{dt}\log S(t\mid \mathbf{x}),$$0 and bounded spline coefficients (Logothetis et al., 21 May 2026).

The main theorem gives the covariate-averaged Hellinger rate

$$\lambda(t\mid \mathbf{x})=-\frac{d}{dt}\log S(t\mid \mathbf{x}),$$1

where $$\lambda(t\mid \mathbf{x})=-\frac{d}{dt}\log S(t\mid \mathbf{x}),$$2 is defined through the conditional law of $$\lambda(t\mid \mathbf{x})=-\frac{d}{dt}\log S(t\mid \mathbf{x}),$$3. The paper emphasizes that the exponent does not depend on covariate dimension. Corollaries yield the same rate, up to the same $$\lambda(t\mid \mathbf{x})=-\frac{d}{dt}\log S(t\mid \mathbf{x}),$$4 factor, for the weighted $$\lambda(t\mid \mathbf{x})=-\frac{d}{dt}\log S(t\mid \mathbf{x}),$$5 error of the log-hazard and for the $$\lambda(t\mid \mathbf{x})=-\frac{d}{dt}\log S(t\mid \mathbf{x}),$$6 error of the implied survival function $$\lambda(t\mid \mathbf{x})=-\frac{d}{dt}\log S(t\mid \mathbf{x}),$$7 on $$\lambda(t\mid \mathbf{x})=-\frac{d}{dt}\log S(t\mid \mathbf{x}),$$8 under censoring positivity assumptions (Logothetis et al., 21 May 2026).

This theory is built using sieve-MLE arguments: spline approximation of KAN edge functions, bracketing entropy control, and likelihood-ratio bounds in bounded supnorm neighborhoods. The key interpretive point is explicit in the paper: the statistical difficulty is governed by the smoothness of the compositional representation rather than by ambient input dimension.

5. Empirical evaluation, benchmarks, and interpretability

KAPLAN-HR is evaluated on six clinical survival benchmarks: METABRIC, RotGBSG, NWTCO, FLCHAIN, SUPPORT, and MIMIC-III. Times are scaled to $$\lambda(t\mid \mathbf{x})=-\frac{d}{dt}\log S(t\mid \mathbf{x}),$$9, covariates are standardized using training statistics, missing rows are dropped, and performance is reported over 10 random train-validation-test splits stratified by event status. Baselines include CoxPH, GAM, HARE, DeepSurv, CoxTime, DeepHit, Deep Survival Machines, CoxKAN, and SuMo-Net (Logothetis et al., 21 May 2026).

The reported metrics are time-dependent concordance, integrated Brier score, integrated calibration index at the median event time, and D-calibration. KAPLAN-HR attains the best time-dependent concordance on four of six datasets and the best IBS on three of six datasets. The only statistically significant superiority after Holm-Bonferroni correction is on FLCHAIN for concordance, where KAPLAN-HR achieves $T\perp C\mid X$0 versus $T\perp C\mid X$1 for the best baseline. On most datasets, 9 or 10 of 10 splits pass the D-calibration test, indicating stable calibration behavior (Logothetis et al., 21 May 2026).

The simulation study is designed to test both approximation theory and interaction modeling. Four synthetic data-generating processes include additive hazards, a non-smooth additive term of smoothness $T\perp C\mid X$2, a covariate interaction term, and a time interaction term. The integrated squared error of the survival estimator decreases with sample size at slopes matching the theoretical rate $T\perp C\mid X$3 for KAPLAN-HR. By contrast, GAM behavior is appropriate on additive DGPs but plateaus on interaction DGPs, which the paper uses to illustrate the value of compositional structure beyond additive spline models (Logothetis et al., 21 May 2026).

Interpretability is retained at the level of edge functions. Because each edge carries a univariate spline-like map, these functions can be plotted directly, and the first-layer decomposition preserves a clear separation between covariate-specific and time-specific contributions before deeper mixing occurs. The paper notes that KANs are suitable for edge visualization and symbolic regression; within KAPLAN-HR, this means that shallow models remain especially close to GAM-style partial-effect analysis, while deeper models still expose their building blocks more transparently than conventional MLP-based survival networks (Logothetis et al., 21 May 2026).

6. Position within survival methodology and terminological boundaries

KAPLAN-HR belongs to the hazard-regression branch of survival modeling rather than to the Kaplan-Meier product-limit tradition. That distinction is important because the term can be confused with Kaplan-Meier or hazard-related inference more broadly. For example, functional central limit theorems for the Kaplan-Meier and Nelson-Aalen estimators under dependent stationary right-censored data concern process limits of product-limit and cumulative-hazard estimators, not joint nonparametric modeling of $T\perp C\mid X$4 by neural architectures (Anevski, 2016). Likewise, bootstrap consistency of the Kaplan-Meier estimator on the whole support addresses Efron resampling for the survival process and its functionals, again in a fundamentally different inferential regime (Dobler, 2016).

Other Kaplan-related developments are also distinct in aim. "Expert Kaplan--Meier estimation" modifies the product-limit estimator under contamination using expert information, either by expert-filtered event indicators or by expert-provided kernels for latent event times (2206.13120). "Sample size calculations for single-arm survival studies using transformations of the Kaplan-Meier estimator" studies fixed-time inference based on transformed Kaplan-Meier estimators, with arcsine square-root and log-minus-log transformations performing better than the classical log approach in small samples (Nagashima et al., 2020). These methods remain centered on product-limit survival estimation or its downstream use, whereas KAPLAN-HR directly parameterizes the conditional hazard as $T\perp C\mid X$5 and estimates $T\perp C\mid X$6 through a KAN architecture (Logothetis et al., 21 May 2026).

The paper also states several limitations. The convergence theory assumes that the true log-hazard is KAN-representable with bounded Sobolev-smooth edge functions, and the asymptotic results do not by themselves characterize finite-sample behavior beyond the reported simulations and benchmarks. The benchmark suite is medium-scale rather than exhaustive. A plausible implication is that KAPLAN-HR is best understood not as a universal replacement for Cox, GAM, or Kaplan-Meier procedures, but as a theoretically structured nonparametric hazard model for settings where interactions and time-varying effects are expected yet additive PH specifications are too restrictive (Logothetis et al., 21 May 2026).