Curve Ergodic Set Regression (CESR)

• CESR is an interpretable model defined by multiplying univariate shape functions to implicitly create interaction terms among features.
• It employs a constrained ESR polynomial basis with normalization, ensuring each univariate curve passes through (0,1) to avoid misleading behaviors.
• CESR serves as a conceptual bridge to Multiplicative-Additive Constrained Models (MACMs) despite limitations from coefficient entanglement.

Curve Ergodic Set Regression (CESR) is an interpretable multiplicative constrained model derived from Ergodic Set Regression (ESR). Its defining construction is the product of univariate feature-level shape functions, which preserves direct curve visualization while implicitly generating interaction terms among all features. In the literature that introduces it, CESR is presented as conceptually attractive because it combines visualizable univariate functions with full-order interaction structure, but it is also diagnosed as practically limited because the same multiplicative parameterization entangles the coefficients of independent and interaction effects. Reported experiments therefore position CESR as an important intermediate model rather than the strongest predictive choice, and motivate Multiplicative-Additive Constrained Models (MACMs) as a direct extension (Wang, 26 Sep 2025).

1. Formal model and normalization

CESR is built from the ESR polynomial basis

$$\left(1+x_{1}^{1}+\cdots+x_{1}^{n_{1}}\right)\cdot \left(1+x_{2}^{1}+\cdots+x_{2}^{n_{2}}\right)\cdot \cdots\cdot \left(1+x_{k}^{1}+\cdots+x_{k}^{n_{k}}\right),$$

which generates a set of power terms $[E_N(X)]$ containing both univariate powers and cross-products. CESR is then defined as a constrained version of ESR with nonlinear parameterization: $$\begin{aligned} CESR&:[W_{nonli}] \times [E_N(X)]^T \ &= C \cdot \prod_{i=1}^{k}\left(1+w_{i1}x_i^{1}+\cdots+w_{in_i}x_i^{n_i}\right) \ &= C \cdot U_1(x_1)\cdot U_2(x_2)\cdot \cdots \cdot U_k(x_k), \end{aligned}$$ where $U_i(x_i)=1+w_{i1}x_i^1+\cdots+w_{in_i}x_i^{n_i}$ is the univariate shape function for feature $x_i$, $w_{ij}$ is the coefficient on the $j$-th power of feature $i$, $n_i$ is the polynomial order for feature $x_i$, and $[E_N(X)]$0 is a global normalization constant (Wang, 26 Sep 2025).

The model prediction is therefore

$[E_N(X)]$1

This is the defining structural property of CESR: it is a multiplicative model whose apparent low-dimensionality comes from representing each feature through a univariate curve and combining those curves through multiplication rather than summation.

A distinctive normalization property is that each $[E_N(X)]$2 passes through $[E_N(X)]$3. The stated reason is that the bias term is extracted outside the individual shape functions, so the resulting constant $[E_N(X)]$4 becomes the product of all those biases. The paper associates this normalization with avoiding misleading behavior and removing infinitely many equivalent local minima. In that sense, CESR is not merely a product of arbitrary univariate polynomials; it is a constrained product with explicitly normalized feature curves.

2. Representational structure and interpretability

When expanded, CESR contains a constant term, univariate terms such as $[E_N(X)]$5, and higher-order interaction terms such as $[E_N(X)]$6, $[E_N(X)]$7, up to products over all features. The paper treats this expansion as central to CESR’s meaning: independent feature effects are represented by the univariate terms inside each $[E_N(X)]$8, while interaction effects arise implicitly through multiplication across different $[E_N(X)]$9 (Wang, 26 Sep 2025).

This representational scheme produces a particular form of interpretability. Each $$\begin{aligned} CESR&:[W_{nonli}] \times [E_N(X)]^T \ &= C \cdot \prod_{i=1}^{k}\left(1+w_{i1}x_i^{1}+\cdots+w_{in_i}x_i^{n_i}\right) \ &= C \cdot U_1(x_1)\cdot U_2(x_2)\cdot \cdots \cdot U_k(x_k), \end{aligned}$$0 can be visualized as a curve, so CESR retains the familiar shape-function view associated with interpretable additive models. At the same time, the final prediction is the product of those curves, scaled by $$\begin{aligned} CESR&:[W_{nonli}] \times [E_N(X)]^T \ &= C \cdot \prod_{i=1}^{k}\left(1+w_{i1}x_i^{1}+\cdots+w_{in_i}x_i^{n_i}\right) \ &= C \cdot U_1(x_1)\cdot U_2(x_2)\cdot \cdots \cdot U_k(x_k), \end{aligned}$$1. The paper therefore describes CESR as showing the “multiplicative effect of all factors on the prediction.”

The interpretability is thus direct at the level of feature-wise transformation but indirect at the level of contribution decomposition. CESR does not have separately parameterized main-effect curves and interaction curves. Instead, a plotted curve $$\begin{aligned} CESR&:[W_{nonli}] \times [E_N(X)]^T \ &= C \cdot \prod_{i=1}^{k}\left(1+w_{i1}x_i^{1}+\cdots+w_{in_i}x_i^{n_i}\right) \ &= C \cdot U_1(x_1)\cdot U_2(x_2)\cdot \cdots \cdot U_k(x_k), \end{aligned}$$2 is interpretable as a univariate component of the model, while its realized effect on prediction is context-dependent because it is multiplied by the other feature factors. A feature’s effective contribution is therefore conditional on the values of the remaining multiplicative terms.

This structure differentiates CESR from models that expose pairwise or higher-order interaction functions explicitly. In CESR, interaction is latent in the algebra of the product. The visual object is still one curve per feature, but the semantics of those curves differ from additive contribution plots.

3. Relation to GAMs and the coefficient-entanglement problem

The comparison class emphasized in the paper is the generalized additive model,

$$\begin{aligned} CESR&:[W_{nonli}] \times [E_N(X)]^T \ &= C \cdot \prod_{i=1}^{k}\left(1+w_{i1}x_i^{1}+\cdots+w_{in_i}x_i^{n_i}\right) \ &= C \cdot U_1(x_1)\cdot U_2(x_2)\cdot \cdots \cdot U_k(x_k), \end{aligned}$$3

where $$\begin{aligned} CESR&:[W_{nonli}] \times [E_N(X)]^T \ &= C \cdot \prod_{i=1}^{k}\left(1+w_{i1}x_i^{1}+\cdots+w_{in_i}x_i^{n_i}\right) \ &= C \cdot U_1(x_1)\cdot U_2(x_2)\cdot \cdots \cdot U_k(x_k), \end{aligned}$$4 is the link function, $$\begin{aligned} CESR&:[W_{nonli}] \times [E_N(X)]^T \ &= C \cdot \prod_{i=1}^{k}\left(1+w_{i1}x_i^{1}+\cdots+w_{in_i}x_i^{n_i}\right) \ &= C \cdot U_1(x_1)\cdot U_2(x_2)\cdot \cdots \cdot U_k(x_k), \end{aligned}$$5 is an intercept, and each $$\begin{aligned} CESR&:[W_{nonli}] \times [E_N(X)]^T \ &= C \cdot \prod_{i=1}^{k}\left(1+w_{i1}x_i^{1}+\cdots+w_{in_i}x_i^{n_i}\right) \ &= C \cdot U_1(x_1)\cdot U_2(x_2)\cdot \cdots \cdot U_k(x_k), \end{aligned}$$6 is a univariate shape function. Both GAMs and CESR offer interpretable univariate feature functions. The difference is structural: GAMs combine them additively, whereas CESR combines them multiplicatively (Wang, 26 Sep 2025).

This structural difference gives CESR a formal interaction advantage. GAMs include only additive independent effects, and GA$$\begin{aligned} CESR&:[W_{nonli}] \times [E_N(X)]^T \ &= C \cdot \prod_{i=1}^{k}\left(1+w_{i1}x_i^{1}+\cdots+w_{in_i}x_i^{n_i}\right) \ &= C \cdot U_1(x_1)\cdot U_2(x_2)\cdot \cdots \cdot U_k(x_k), \end{aligned}$$7Ms extend GAMs with pairwise interactions but not beyond pairwise. CESR, by contrast, implicitly contains interactions of all orders among the features because the product of univariate shape functions expands into all corresponding cross-terms.

The paper’s central negative result is that symbolic inclusion of more term types does not imply a broader useful hypothesis space. In CESR, coefficients of independent terms and interaction terms are intertwined. A coefficient such as $$\begin{aligned} CESR&:[W_{nonli}] \times [E_N(X)]^T \ &= C \cdot \prod_{i=1}^{k}\left(1+w_{i1}x_i^{1}+\cdots+w_{in_i}x_i^{n_i}\right) \ &= C \cdot U_1(x_1)\cdot U_2(x_2)\cdot \cdots \cdot U_k(x_k), \end{aligned}$$8 is simultaneously the coefficient of the independent term $$\begin{aligned} CESR&:[W_{nonli}] \times [E_N(X)]^T \ &= C \cdot \prod_{i=1}^{k}\left(1+w_{i1}x_i^{1}+\cdots+w_{in_i}x_i^{n_i}\right) \ &= C \cdot U_1(x_1)\cdot U_2(x_2)\cdot \cdots \cdot U_k(x_k), \end{aligned}$$9 and a multiplicative factor inside the coefficients of interaction terms involving $U_i(x_i)=1+w_{i1}x_i^1+\cdots+w_{in_i}x_i^{n_i}$0. Adjusting a marginal effect therefore automatically changes multiple interaction coefficients. The paper states this directly: “Such coupled coefficients prevent the hypothesis space of CESR from encompassing that of GAMs.”

The concrete example given is

$U_i(x_i)=1+w_{i1}x_i^1+\cdots+w_{in_i}x_i^{n_i}$1

If one tries to represent this using only the multiplicative part

$U_i(x_i)=1+w_{i1}x_i^1+\cdots+w_{in_i}x_i^{n_i}$2

then fitting the constant, $U_i(x_i)=1+w_{i1}x_i^1+\cdots+w_{in_i}x_i^{n_i}$3, and $U_i(x_i)=1+w_{i1}x_i^1+\cdots+w_{in_i}x_i^{n_i}$4 terms exactly yields

$U_i(x_i)=1+w_{i1}x_i^1+\cdots+w_{in_i}x_i^{n_i}$5

These imply

$U_i(x_i)=1+w_{i1}x_i^1+\cdots+w_{in_i}x_i^{n_i}$6

so the multiplicative expansion becomes

$U_i(x_i)=1+w_{i1}x_i^1+\cdots+w_{in_i}x_i^{n_i}$7

which misses the extra $U_i(x_i)=1+w_{i1}x_i^1+\cdots+w_{in_i}x_i^{n_i}$8 term. This is the paper’s clearest demonstration that CESR can include both $U_i(x_i)=1+w_{i1}x_i^1+\cdots+w_{in_i}x_i^{n_i}$9 and $x_i$0 structurally while still failing to assign them independently chosen coefficients.

4. Empirical behavior and comparative performance

The reported experiments do not show CESR outperforming GAM-based or newer shape-function models. On the Pima Indian Diabetes dataset, the paper reports CESR at $x_i$1 AUC and GAMs(Poly) at $x_i$2 AUC, which the authors use as an early indication that CESR does not beat the additive baseline (Wang, 26 Sep 2025).

In the main experiments, CESR performs worse than NAMs, NBMs, ProtoNAM, and MACMs on all listed tasks. The paper reports the following CESR results, alongside MACMs(NNs) for scale:

Dataset	CESR	MACMs(NNs)
CA Housing (modified)	$x_i$3 RMSE	$x_i$4
CA Housing	$x_i$5 RMSE	$x_i$6
Stroke	$x_i$7 AUC	$x_i$8
Water Quality	$x_i$9 RMSE	$w_{ij}$0

The intermediate polynomial extension also improves over CESR. MACMs(poly) is reported as $w_{ij}$1 on CA Housing (modified), $w_{ij}$2 on Stroke, and $w_{ij}$3 on Water Quality, all better than the corresponding CESR values. The ablation study further reports that both multiplicative-only and additive-only versions underperform the full MACM, which the authors interpret as evidence that both components are necessary.

Taken together, these results support the paper’s main empirical judgment: CESR is more interesting conceptually than competitive practically. It shows how visualizable univariate curves can coexist with full-order interactions, but its entangled coefficients limit predictive performance.

5. MACM as the direct extension of CESR

The proposed extension is the Multiplicative-Additive Constrained Model, first given in polynomial form as

$w_{ij}$4

and then generalized as

$w_{ij}$5

Within this formulation, the multiplicative part continues to generate interaction terms, while the additive part introduces free coefficients for independent terms (Wang, 26 Sep 2025).

The key purpose of MACM is coefficient disentanglement. In polynomial language, interaction coefficients still arise from products such as $w_{ij}$6, but independent terms now have coefficients such as $w_{ij}$7. This allows main effects to be adjusted without forcing simultaneous changes in interaction coefficients. The earlier counterexample is then repaired by adding an appropriate additive component.

The paper also provides a hypothesis-space argument. It defines a multiplicative hypothesis space $w_{ij}$8 and an additive hypothesis space $w_{ij}$9, and argues that when $j$0, these spaces have little intersection, so that

$j$1

The stated implication is that MACM broadens the function class by combining two largely distinct hypothesis spaces rather than appending a redundant correction.

Interpretability is preserved, but modified. After normalization, the model is written as

$j$2

where $j$3 and $j$4 has no bias term. The paper further isolates the contribution of one feature as

$j$5

with

$j$6

This yields what the authors call a “semi-global” interpretability story: independent and interactive mechanisms can be visualized separately, but the effective contribution of one feature still depends on the context through $j$7.

6. Scope, implementation boundaries, and neighboring literatures

For CESR itself, the paper does not provide a detailed modern neural training recipe. It presents CESR mainly in polynomial and constrained functional form, and states that optimizer, architectural depth, and regularization are not fully defined beyond that parameterization (Wang, 26 Sep 2025). By contrast, MACMs(poly) and MACMs(NNs) are specified more concretely, including degree $j$8, scaling factor $j$9, Adam, batch size $i$0, learning rate $i$1, and $i$2 epochs for MACMs(poly), and fully connected shape-function subnetworks with $i$3 hidden layers, $i$4 neurons per layer, ReLU activations, min-max normalization to $i$5, and separate regression and binary-classification settings for MACMs(NNs). This asymmetry in implementation detail is itself informative: CESR is presented primarily as a model family and conceptual precursor.

The paper also states explicit trade-offs that remain even after extending CESR. Greater complexity reduces transparency compared with classical GAMs, dynamic shape functions provide only semi-global interpretability, and the multiplicative branch is sensitive to the number of features and requires careful tuning of the scaling factor $i$6. These caveats apply even more strongly to CESR, whose entire mechanism is multiplicative.

A recurring source of confusion is nomenclature. In accelerator physics, “CESR” denotes the Cornell Electron Storage Ring rather than Curve Ergodic Set Regression (Bergan et al., 2017). Likewise, stationary-ergodic nonparametric regression without mixing assumptions (0712.2592) and right-censored kernel $i$7-estimation for stationary ergodic data (Chaouch et al., 2016) study “ergodic regression” in a time-series sense, but do not define CESR as an interpretable multiplicative shape-function model. Regression for curve-valued responses in quotient metric spaces, especially elastic curves modulo re-parameterization (Steyer et al., 2023), addresses a different notion of “curve regression,” and switching nonparametric regression for multi-curve data with latent Markov states (Souza et al., 2015) addresses yet another distinct framework. These neighboring literatures are relevant for comparison of terminology, but the CESR introduced in interpretable machine learning remains specifically the multiplicative constrained model derived from ESR and critiqued through the coefficient-entanglement argument.

In that sense, CESR occupies a precise place in the interpretable-model landscape. It demonstrates that an interpretable model need not be purely additive, because multiplying univariate shape functions automatically introduces higher-order interactions. At the same time, the literature introducing CESR argues that this very construction restricts the usable hypothesis space, so the model’s lasting significance is as a bridge: it exposes a non-additive route to interpretable shape-function modeling, and directly motivates the multiplicative-additive designs that follow.