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REVIEW 4 major objections 5 minor 46 references

Forecasting VIX using interpretable Kolmogorov-Arnold networks

T0 review · 4 major / 5 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read The paper claims that a minimal Kolmogorov-Arnold network turns VIX forecasting into a closed-form linear equation that matches much larger neural networks and exposes mean reversion and the leverage effect.

desk verdict A competent empirical KAN-for-VIX study with believable forecasting results, but the mean-reversion "insight" is a relabeling of the fitted line, not a finding. read the letter →

arxiv 2502.00980 v1 pith:CHSJT4V5 submitted 2025-02-03 cs.LG cs.AIcs.CE

So-Yoon Cho , Sungchul Lee , Hyun-Gyoon Kim This is my paper · ORCID
classification cs.LGcs.AIcs.CE
keywords Kolmogorov-ArnoldnetworksVIXforecastinginterpretablemachinelearningmeanreversionleverageeffectsymbolificationtime-seriesvolatility
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

VIX forecasting usually forces a choice between accurate but opaque neural networks and simple but restrictive statistical models. This paper claims that a pure Kolmogorov-Arnold network (KAN)—a two-layer network with just two hidden nodes and spline activations on its edges—sits in the middle: after training, its learned activations are almost exactly linear, so the whole forecast can be written out as a closed-form linear equation in lagged VIX inputs. Across three predictor sets and three train/validation/test splits, these tiny networks match or beat MLP and LSTM benchmarks while using roughly 300 to 1,000 times fewer parameters, and the symbolified formulas reproduce known VIX facts: persistence through a dominant $V_{t-1}$ coefficient, pruning of long-horizon averages, and a negative coefficient on the previous day's S&P 500 excess return (the leverage effect). The paper further contends that the Dataset 3 formulas rearrange into a mean-reverting process of the form $\kappa(\theta - V_{t-1}) + \varepsilon_t$.

What carries the argument

The object that carries the argument is the Kolmogorov-Arnold network, a neural network whose trainable parameters are univariate spline activation functions on the edges rather than linear weights, with the Kolmogorov-Arnold representation theorem as its theoretical basis. Here it takes a two-layer form with two hidden nodes, trained on squared error, then pruned and symbolified—the learned splines are replaced by the closest elementary functions, here $y=x$ and $y=0$, followed by affine fine-tuning. The mechanism that matters is that the trained splines are nearly linear, so the whole network collapses into a closed-form linear equation; pruning also drops irrelevant inputs such as $V_{t-21}$ and the quarterly average $V_q$, turning the model into an interpretable variable-selection device.

What would settle it

Take the Dataset 3 closed-form expression for any period, compute the residual $\hat{\varepsilon}_t = \Delta V_t - \hat{\kappa}(\hat{\theta} - V_{t-1})$ from the fitted coefficients, then regress $\hat{\varepsilon}_t$ on $V_{t-1}$ or bin $V_{t-1}$ and test the mean within each bin; if the conditional mean of $\hat{\varepsilon}_t$ is not zero across levels of $V_{t-1}$, the mean-reversion interpretation in equation (3.4) does not follow from the fitted model.

Watch

Extended reading notes

Core claim

The paper's central claim is that a deliberately minimal KAN is a competitive, interpretable VIX forecaster. Using only lagged VIX values (five consecutive days, four non-uniform lags, or yesterday's VIX plus weekly/monthly/quarterly averages), a two-layer KAN with two hidden nodes is trained out-of-sample; the B-spline activations consistently train to near-linear shapes, and after pruning and symbolification the network reduces to explicit formulas such as $\hat{V}_t = 0.8584 V_{t-1} + 0.0206 V_{t-2} + 0.0632 V_{t-3} - 0.0374 V_{t-4} + 0.0801 V_{t-5} + 0.2793$. These formulas match the forecasting accuracy of MLPs with 20,901 parameters and LSTMs with roughly 63,500 to 63,800 parameters while using only 60 to 72 parameters, and symbolification improves $R^2$ in eight of nine setups. The paper interprets the Dataset 3 formulas as a typical mean-reverting process and, after augmenting the model with the lagged S&P 500 excess return, as evidence of the leverage effect.

Load-bearing premise

The load-bearing premise is that the leftover term $\varepsilon_t$ in equation (3.4) can be treated as zero-mean noise even though it is allowed to depend on $V_{t-1}$; if that premise fails, the mean-reversion reading of the fitted equation is just a relabeling.

Editorial extensions

If this is right

  • VIX forecasting can be delivered as a transparent equation rather than a black-box prediction, with explicit coefficients for each lag and average.
  • A very small KAN (60 to 72 parameters) is enough to reach the accuracy of MLPs and LSTMs that are hundreds to thousands of times larger, so interpretability does not have to come at the cost of accuracy.
  • Symbolification slightly improves out-of-sample $R^2$ in eight of the nine data/period combinations, suggesting that parsimonious linear structure helps in noisy financial time series.
  • The lagged S&P 500 excess return systematically enters with a negative coefficient except in one anomalous setup, giving a learned closed-form expression of the leverage effect.
  • For one-day-ahead forecasts, very long-horizon averages are pruned, so the useful memory lies in the most recent VIX level and short/medium-term averages rather than monthly or quarterly averages.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Because every trained spline is near-linear, the KAN is effectively discovering a linear model in this application; the practical gain is data-driven lag selection and a closed-form output rather than a nonlinear functional form.
  • A direct test of whether the residual in equation (3.4) has zero mean conditional on yesterday's VIX would turn the mean-reversion reading from a rearrangement of the fitted line into an empirical finding.
  • The same pruning-and-symbolification recipe could be applied to other persistent volatility series, where a natural check is whether the linear activations and the specific pruned horizons recur or whether nonlinear forms appear.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. The paper applies two-layer Kolmogorov-Arnold Networks (KANs) to one-day-ahead VIX forecasting, using three lag/aggregate datasets and three train–validation–test splits. After pruning and symbolification, the KAN collapses to closed-form linear forecasting equations. The paper reports that these KANs achieve accuracy comparable to or slightly better than MLP, LSTM, ARMA/ARIMA, and HAR baselines while using far fewer parameters, and it claims that the closed forms reveal two interpretable facts about the VIX: mean reversion and the leverage effect.

Significance. If the empirical claims hold, the paper is a useful contribution to interpretable financial machine learning: it demonstrates that a very small symbolic KAN can match the out-of-sample accuracy of much larger neural networks on a standard volatility-forecasting task, and the closed-form output is genuinely transparent. The paper's strengths include a systematic comparison across nine configurations, an honest report that symbolification usually slightly improves R-squared, and a candid acknowledgment that some symbolification replacements for the leverage effect are not strictly accurate. However, the mean-reversion interpretation in Section 3.2.2 is not established by the derivation given, the forecasting comparisons lack any uncertainty quantification, and a few coefficient inconsistencies need correction. These issues affect the paper's advertised interpretability contribution rather than the raw forecasting tables.

major comments (4)
  1. [3.2.2, Eqs. (3.3)–(3.4), Table 2] The mean-reversion interpretation is not supported. First, Eq. (3.3) uses coefficients (0.8290, 0.1472, 0.4866) that do not match the D3P1 closed form in Table 2 (0.8297, 0.1477, 0.4756), so the calculation is based on an expression that is not the fitted model. Second, defining epsilon_t as whatever remains after extracting kappa(theta - V_{t-1}) makes Eq. (3.4) a relabeling: the epsilon_t in this case is -0.0238V_{t-1}+0.4866, whose conditional expectation depends on V_{t-1}, so it is not a zero-mean innovation in any standard sense. The claim that the KAN-derived equation 'can be interpreted as representing a typical mean-reverting process' therefore does not follow. Since the abstract and conclusion advertise mean reversion as a key interpretable insight, this point should be either substantiated (e.g., by testing E[epsilon_t | V_{t-1}] = 0 and showing the decomposition is not purely arbitrary) or removed from the claims.
  2. [3.2.5, Tables 5–7] All forecasting comparisons are point estimates without error bars, confidence intervals, or significance tests. The differences between KAN and the HAR or forward-filling baselines are very small (for example, Period 3 MSE is 1.9575 for KAN-D3 versus 1.9579 for HAR(4), and Period 1 MSE is 4.3957 for KAN-D3 versus 4.4337 for HAR(3)), so the statement that KAN 'tends to excel in MSE' is not statistically supported. The paper should report standard errors, Diebold-Mariano tests, or at least multiple random seeds with mean +/- std. In addition, the '# params' column should state exactly which parameters are counted; if it counts only the affine parameters after symbolification, the comparison with MLP/LSTM parameter counts is not a like-for-like measure of model complexity during training.
  3. [3.2.1, ARMA/ARIMA order selection] The order selection for ARMA and ARIMA is described only as 'the optimal values of p and q are selected based on AIC' and 'd is determined using the KPSS test,' without specifying whether this is done on the training split of each period or on the full sample. If the same ARMA(1,1) and ARIMA(1,1,1) specifications are used for all three periods after a single in-sample fit, those baselines are not strictly out-of-sample, and the comparison in Tables 5–7 is unfair to the neural networks, which are refitted per period. Please clarify the selection procedure and, if necessary, redo the baseline estimation within each training window.
  4. [3.2.4, Table 4] The quantitative leverage-effect coefficients in Table 4 are obtained by manually replacing the learned activation function for R^e_{t-1} with y = -x, which the authors acknowledge is not strictly accurate. The reported coefficients therefore describe a constrained auxiliary model, not the fitted KAN. The visual downward slope in Figure 2 is suggestive, but the paper should either quantify the approximation error of the y = -x replacement or present the leverage-effect conclusion as qualitative only. The current wording ('the coefficients for R^e_{t-1} range from -0.02 to -0.07... this result suggests a nonnegligible negative impact') overstates the evidential weight.
minor comments (5)
  1. [Abstract and Section 3.2.2] There are several typos and spacing errors: 'V olatility' in the abstract, 'Moverover' and 'the the value' in Section 3.2.2, and the repeated 'a ffine' should all be corrected to 'affine'.
  2. [2.2] The layer definition uses q for the input index and p for the output index, which is the reverse of the convention used in the original KAN paper [26]; consider aligning the notation to avoid confusion.
  3. [3.2.2] The text says 'the coefficient of Vt-1 is estimated at 0.8290' while Table 2 gives 0.8297 for D3P1; this is part of the inconsistency noted in Major Comment 1 and should be corrected independently.
  4. [3.2.3] It should be stated explicitly whether the Mincer-Zarnowitz and Durbin-Watson tests are applied to out-of-sample forecast errors; currently this is only implicit.
  5. [Data availability] The data availability statement is vague ('available from DataGuide... restrictions apply'). Please add a code availability statement or a detailed description of the KAN implementation (library version, hyperparameters, pruning thresholds) to enable reproducibility.

Circularity Check

1 steps flagged · score 6.0 of 10

Mean-reversion insight is a relabeling: Eq. (3.4) is made true by defining ε_t as the V_{t-1}-dependent residual, so this advertised interpretable result is not empirically demonstrated; the forecasting comparisons themselves remain independent.

  1. self definitional [Section 3.2.2, Eqs. (3.3)–(3.4)]
    "If we let (−0.0238Vt−1 + 0.4866) in Period 1 be ϵt, then the equation (3.3) becomes identical to (3.4). Consequently, the closed-form expressions derived by KAN with Dataset 3 can be interpreted as representing a typical mean-reverting process reverting to its medium- or long-term average."

    Eq. (3.4) is imposed rather than tested: the paper defines ϵt as the leftover term, explicitly allowing ϵt to depend on Vt−1. Since the residual −0.0238Vt−1 + 0.4866 is a function of Vt−1, E[ϵt|Vt−1] ≠ 0, so the conditional expectation of ΔVt is not κ(θ − Vt−1). The actual Vt−1 coefficient in (3.3) is −0.1472 − 0.0238 = −0.171, not κ = 0.1472, and the 'mean level' θ would have to be the time-varying weekly average Vw rather than the constant θ in (3.4). Any persistent linear autoregression of this form can be rewritten as (3.4) by absorbing Vt−1 dependence into ϵt, so the claimed empirical demonstration of mean reversion is a relabeling, not a derivation.

full rationale

The forecasting-performance core of the paper is a self-contained benchmark comparison: KAN is compared against forward filling, ARMA, ARIMA, HAR, MLP, and LSTM on out-of-sample data in Tables 5–7, and the parameter-count comparison is direct. Those claims are not circular. However, the paper's other advertised headline insight, mean reversion, is circular: Eq. (3.4) is satisfied by construction because the paper permits ϵt to depend on Vt−1 and defines it as the residual term in Eq. (3.3). The 'typical mean-reverting process' conclusion is therefore a relabeling of the fitted linear equation rather than an empirical finding. Because this is a central interpretability contribution, though not the forecasting comparison itself, the circularity is partial rather than total.

Assumptions & free parameters 14 free parameters · 6 assumptions · 0 invented entities

No new physical or mathematical entities are introduced. The central claim depends on fitted regression coefficients, standard approximation theorems, and several domain assumptions about the VIX process. The most fragile assumptions are the symbolification candidate set and the mean-reversion noise specification.

free parameters (14)
  • D1P1 regression coefficients = 0.8482*Vt-1 + 0.0536*Vt-2 + ... + 0.3779
    Fitted on training data; gives the central forecast formula for Dataset 1, Period 1.
  • D1P2 regression coefficients = 0.8785*Vt-1 + ... + 0.2345
    Fitted on training data for Dataset 1, Period 2.
  • D1P3 regression coefficients = 0.8584*Vt-1 + ... + 0.2793
    Fitted on training data for Dataset 1, Period 3.
  • D2P1 regression coefficients = 0.8661*Vt-1 + 0.0634*Vt-5 + 0.0655*Vt-10 + 0.0712
    Fitted on training data for Dataset 2, Period 1.
  • D2P2 regression coefficients = 0.8800*Vt-1 + 0.0733*Vt-5 + 0.0359*Vt-10 + 0.1950
    Fitted on training data for Dataset 2, Period 2.
  • D2P3 regression coefficients = 0.8840*Vt-1 + 0.0742*Vt-5 + 0.0344*Vt-10 + 0.1099
    Fitted on training data for Dataset 2, Period 3.
  • D3P1 regression coefficients = 0.8297*Vt-1 + 0.1477*Vw + 0.4756
    Fitted on training data for Dataset 3, Period 1.
  • D3P2 regression coefficients = 0.8189*Vt-1 + 0.1660*Vw + 0.2749
    Fitted on training data for Dataset 3, Period 2.
  • D3P3 regression coefficients = 0.8255*Vt-1 + 0.1292*Vw + 0.0377*Vm + 0.1025
    Fitted on training data for Dataset 3, Period 3.
  • Leverage model affine coefficients (Table 4) = e.g., 0.9974*Vhat - 0.0662*R_{t-1} - 0.0393 for D1P1
    Fitted in the second-stage KAN with excess returns; used to claim the leverage effect.
  • B-spline grid size = 3
    Hand-chosen hyperparameter for activation function resolution.
  • B-spline order = 3
    Hand-chosen hyperparameter for spline basis.
  • Regularization coefficient lambda = 0
    Set to zero during training; pruning is done by importance threshold instead.
  • Learning rates and early stopping = 0.04 then 0.0004
    Hand-chosen optimization settings; not part of the economic claim.
assumptions (6)
  • standard math Kolmogorov-Arnold representation theorem
    Invoked in Section 2.1 to justify representing multivariate functions with univariate activations.
  • standard math B-spline basis functions can approximate the learned activations
    Standard approximation result; used throughout the KAN training and symbolification.
  • domain assumption The VIX is a function of the chosen explanatory variables plus noise, and out-of-sample evaluation reflects predictive ability
    Underlies the entire forecasting setup; not proven, but standard in time-series forecasting.
  • ad hoc to paper Symbolification with only y=x and y=0 is sufficient to represent the true relationship
    In Section 3.2.2 the authors restrict candidate symbolic functions to y=x and y=0 after observing near-linear shapes; this choice is not justified as exhaustive.
  • domain assumption The residual from the VIX-only model separates additively into a function of excess returns plus irreducible noise
    Eq. (3.7) assumes V_t = Vhat_t + psi(R_{t-1}) + e_t, which is not derived from the trained model.
  • ad hoc to paper epsilon_t in Eq. (3.4) can have zero mean while depending on V_{t-1}
    This assumption makes the mean-reversion representation hold by construction, but it is not a standard noise assumption for mean-reverting processes.

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Cite this review

Pith. "Pith review of Forecasting VIX using interpretable Kolmogorov-Arnold networks." pith.science (2026). https://pith.science/paper/CHSJT4V5

@misc{pith2026250200980,
  author       = {Pith},
  title        = {Pith review of: Forecasting VIX using interpretable Kolmogorov-Arnold networks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CHSJT4V5}},
  note         = {Machine review of arXiv:2502.00980}
}
read the original abstract

This paper presents the use of Kolmogorov-Arnold Networks (KANs) for forecasting the CBOE Volatility Index (VIX). Unlike traditional MLP-based neural networks that are often criticized for their black-box nature, KAN offers an interpretable approach via learnable spline-based activation functions and symbolification. Based on a parsimonious architecture with symbolic functions, KAN expresses a forecast of the VIX as a closed-form in terms of explanatory variables, and provide interpretable insights into key characteristics of the VIX, including mean reversion and the leverage effect. Through in-depth empirical analysis across multiple datasets and periods, we show that KANs achieve competitive forecasting performance while requiring significantly fewer parameters compared to MLP-based neural network models. Our findings demonstrate the capacity and potential of KAN as an interpretable financial time-series forecasting method.

Figures

Figures reproduced from arXiv: 2502.00980 by the authors.

Figure 1
Figure 1. KAN training results for Datasets 1–3 under Period 3 before (top row) and after (bottom row) symbolification. Black activation functions indicate the trained B-spline-based structure, whereas red activation functions depict the replaced symbolic functions. Vivid edges represent stronger effects, while fainter edges mean weaker ones. 8 [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. Training results of KAN(Vˆ t , R e t−1 ) across Datasets and Periods. To clearly see the activation functions, the contrast of the figures is increased by three times compared to [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗

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Reviewed August 9, 2026 · model on record in the stance chip above.