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KAPLAN: Kolmogorov-Arnold Prognostic Learnable Activation Networks for Survival Analysis

The paper introduces KAPLAN-HR, a B-spline Kolmogorov-Arnold Network for survival analysis that automatically learns complex covariate interactions and time-varying effects without manual specification, offering dimension-independent convergence rates and superior predictive performance across clinical datasets.

Original authors: Stelios Boulitsakis Logothetis, Angela Wood, Pietro Li ò

Published 2026-05-25

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Stelios Boulitsakis Logothetis, Angela Wood, Pietro Li ò

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are a doctor trying to predict how long a patient will live after a diagnosis, or a mechanic trying to guess when a car part will break. This is called Survival Analysis. The tricky part is that you often don't get to see the full story. Sometimes a patient moves away, or a car gets sold before it breaks. You only know they survived at least until that point. This is called "right censoring."

For decades, statisticians have used tools like the Cox model to make these predictions. Think of the Cox model as a very rigid, pre-fabricated house. It's sturdy and easy to understand, but if you want to add a spiral staircase or a skylight (complex interactions between variables), you have to manually design and build it yourself. If you miss a detail, the whole house might not fit the reality of the data.

Enter KAPLAN-HR, the new method introduced in this paper. The authors (from the University of Cambridge) propose a smarter, more flexible way to build these prediction models using something called Kolmogorov-Arnold Networks (KANs).

Here is how KAPLAN-HR works, explained through simple analogies:

1. The "Lego" vs. The "Puzzle"

Traditional deep learning models (like standard neural networks) are like a giant, solid block of concrete. They are powerful and can learn almost anything, but they are hard to understand. You can't easily see why they made a specific prediction.

Older statistical models (like GAMs) are like a set of Lego bricks. You can snap them together, but each brick only does one simple thing (like "how age affects risk" or "how time affects risk"). If you want to see how age and time together affect risk in a complex way, you have to manually tell the model to snap those specific bricks together.

KAPLAN-HR is like a smart, self-assembling puzzle.

The Edges: Instead of fixed numbers, the connections in the network are "learnable functions." Imagine each connection is a flexible rubber band that can stretch and twist into any shape it needs to be to fit the data.
The Layers:
- One Layer: If you use just one layer, it acts like the old Lego models (GAMs). It's simple, additive, and easy to interpret.
- Multiple Layers: If you stack more layers, the rubber bands start interacting. The model can now figure out complex relationships on its own—like how a specific drug might only work if a patient is also over a certain age, without you having to tell it to look for that specific combination.

2. The "Curse of Dimensionality" Breaker

In data science, there's a problem called the "curse of dimensionality." It's like trying to find a specific needle in a haystack that keeps getting bigger and bigger. Usually, as you add more variables (covariates) to your model, the amount of data you need to get a good answer explodes.

The paper claims that KAPLAN-HR has a special superpower: it ignores the size of the haystack.
The authors proved mathematically that if the real-world truth can be described by this "flexible puzzle" structure, the model learns just as fast whether you have 5 variables or 500. It doesn't get slower or need more data just because the problem got more complex. This is a huge deal because medical datasets often have hundreds of different measurements.

3. The "Training" Process

How does the model learn?

It looks at the data (patients, times, and whether the event happened).
It tries to draw a curve (the "hazard") that fits the data best.
Because it uses B-splines (a type of smooth mathematical curve) combined with a little bit of a modern activation function (SiLU), it can wiggle and bend to fit the data perfectly without being too jagged or too smooth.
It uses a "numerical integration" trick to turn its predictions into survival curves (the probability of surviving to a certain time), which is necessary because the math gets too messy to solve with a simple formula.

4. Did it Work? (The Results)

The authors tested KAPLAN-HR on six real-world clinical datasets (including breast cancer data, heart failure data, and ICU patient records). They compared it against:

The old standard (Cox models).
The flexible Lego models (GAMs).
The big concrete blocks (Deep Learning models like DeepSurv and DeepHit).

The Verdict:
KAPLAN-HR was a champion.

It matched or beat the best existing models in predicting who would survive longer (Discrimination).
It was very accurate in predicting the exact survival curve (Accuracy).
It was well-calibrated, meaning if it said there was a 20% chance of an event, it actually happened about 20% of the time.

Crucially, in their simulations, they showed that when the data had complex interactions (like a "twist" in the relationship between variables), the old models got stuck and stopped improving, while KAPLAN-HR kept getting better as it saw more data.

Summary

KAPLAN-HR is a new tool for predicting time-to-event data (like survival times). It bridges the gap between simple, easy-to-understand statistical models and complex, black-box deep learning models. It uses a flexible, puzzle-like architecture that can automatically discover complex relationships in the data without needing a human to manually program them. The paper proves that this method is mathematically sound (it learns fast regardless of data complexity) and practically effective (it performs as well as or better than the current state-of-the-art on real medical data).

Technical Summary: KAPLAN-HR

Problem Statement

Survival analysis aims to model the time-to-event distribution under right censoring, where the exact event time is only partially observed. While the Cox proportional hazards (PH) model is the standard, it requires manual specification of feature interactions and assumes time-invariant covariate effects, which is increasingly impractical for rich clinical datasets. Existing flexible alternatives, such as Generalised Additive Models (GAMs) and HARE, are structurally limited: GAMs require manual interaction terms, and HARE cannot represent higher-order interactions. Conversely, deep learning approaches relax structural restrictions but often sacrifice the interpretability and additive structure of classical models. Furthermore, while recent applications of Kolmogorov-Arnold Networks (KANs) to survival analysis show promise, they lack formal theoretical analysis regarding convergence rates and statistical properties.

Methodology: KAPLAN-HR

The authors introduce KAPLAN-HR (Kolmogorov-Arnold Prognostic Learnable Activation Networks for Survival Analysis), a B-spline KAN designed to estimate the conditional hazard as a joint function of covariates ( $x$ ) and time ( $t$ ).

Architecture

The model parameterizes the log-hazard function $g(x, t)$ using a KAN, where edges carry learnable univariate functions and nodes sum inputs.

Edge Parameterization: Each edge is a weighted sum of a B-spline and a residual SiLU (Sigmoid Linear Unit) term.
Depth and Structure:
- Single-layer ( $D=1$ ): Recovers an additive B-spline GAM under the PH restriction, where $\log \lambda(t|x) = \sum \phi_i(x^{(i)}) + \phi_t(t)$ .
- Deep architectures ( $D \ge 2$ ): Through composition, deeper layers capture covariate interactions and time-varying effects, removing both the additive and proportional hazards restrictions.
Training: The model minimizes the negative log-likelihood (NLL) of the observed data. Since the cumulative hazard $\Lambda(t|x) = \int_0^t \lambda(s|x) ds$ has no closed form for KAN-modeled hazards, the authors approximate it using numerical integration (right-endpoint rectangle rule) over a discrete time grid.

Theoretical Analysis

The paper establishes a convergence rate for the KAPLAN-HR estimator using the method of sieves.

Assumptions: The true log-hazard $g^*$ is assumed to lie within a class of functions representable as a layered composition of smooth univariate functions ( $FKAN_r$ ). Censoring is assumed independent of the event time conditional on covariates.
Convergence Rate: The authors prove that the Hellinger distance between the estimated and true log-hazard converges at a rate of $O_P(n^{-r/(2r+1)}\sqrt{\log n})$ , where $r$ is the smoothness exponent.
Significance of Rate: Crucially, this rate depends only on the smoothness of the representation and the sample size, not on the covariate dimension (up to a logarithmic factor). This suggests KAPLAN-HR mitigates the curse of dimensionality for targets representable by KANs.

Key Contributions

Model Proposal: KAPLAN-HR serves as a bridge between additive spline-based hazard models and deep survival models, where network depth controls the level of structural relaxation (from additive to interactive).
Theoretical Guarantee: The paper provides the first formal convergence rate for a KAN-based survival model, demonstrating dimension independence for KAN-representable targets.
Empirical Validation: Extensive evaluation on six clinical benchmark datasets and simulation studies.

Experimental Results

Simulations

The authors tested four Data Generating Processes (DGPs) with varying smoothness and interaction structures:

Convergence: KAPLAN-HR tracked the predicted convergence rate ( $n^{-r/(2r+1)}$ ) across all DGPs.
Interactions: While GAMs converged on additive DGPs (DGP-1, DGP-2), they plateaued on interaction-heavy DGPs (DGP-3, DGP-4). KAPLAN-HR continued to improve, demonstrating its ability to capture interactions that additive models cannot.

Clinical Benchmarks

KAPLAN-HR was evaluated on six standard datasets (METABRIC, MIMIC-III, RotGBSG, FLCHAIN, NWTCO, SUPPORT) against nine baselines (CoxPH, GAM, HARE, DeepSurv, DeepHit, DSM, CoxKAN, etc.) using discrimination (C-TD), survival accuracy (IBS), and calibration (ICI, D-CAL).

Performance: KAPLAN-HR achieved the best C-TD on 4 of 6 datasets, best IBS on 3 of 6, and best ICI on 2 of 6.
Significance: In a Wilcoxon signed-rank test with Holm-Bonferroni correction, KAPLAN-HR showed a statistically significant improvement in discrimination on the FLCHAIN dataset compared to the best baseline (DeepSurv). No significant degradations were observed in any metric.
Ranking: On average, KAPLAN-HR ranked competitively, often matching or exceeding established statistical and deep learning methods.

Significance and Claims

The paper claims that KAPLAN-HR offers a principled approach to survival analysis that balances flexibility and interpretability.

Theoretical Novelty: The primary contribution is the establishment of a dimension-free convergence rate for KAN-based hazard estimators, a theoretical gap previously unaddressed in KAN survival literature.
Practical Utility: The method successfully captures complex interactions and time-varying effects without manual specification, outperforming or matching state-of-the-art models on real-world clinical data.
Limitations: The authors note that the convergence results rely on the structural assumption that the true log-hazard is KAN-representable, which may not hold in all real-world scenarios. Additionally, the theoretical guarantees describe asymptotic behavior rather than finite-sample performance. The paper also acknowledges known shortcomings in standard survival analysis benchmarking practices.

In conclusion, KAPLAN-HR demonstrates that KANs are well-suited for survival analysis, providing a model that is theoretically grounded, dimensionally robust, and empirically competitive.