Fundamentals

This page introduces the main mathematical ideas behind High-Dimensional Model Representation (HDMR) and Enhanced Multivariate Products Representation (EMPR) in HDMRLib.

Why HDMR and EMPR?

High-dimensional functions and tensors are difficult to analyze directly because the number of possible interactions grows rapidly with dimension. HDMR and EMPR address this by representing a multivariate object through lower-order terms, which makes approximation and interaction analysis more manageable.

High-Dimensional Model Representation (HDMR)

Let f(x) = f(x1, x2, …, xd) be a multivariate function.

HDMR represents f as a hierarchy of lower-order terms:

\[f(\mathbf{x}) = f_0 + \sum_{i=1}^{d} f_i(x_i) + \sum_{1 \le i < j \le d} f_{ij}(x_i, x_j) + \cdots + f_{12\ldots d}(x_1, \ldots, x_d)\]

Here, f0 is a constant term, fi(xi) are first-order terms, and fij(xi, xj) are second-order interaction terms.

The key idea is that many systems are dominated by low-order effects, so a truncated expansion often provides a useful approximation.

Enhanced Multivariate Products Representation (EMPR)

EMPR follows the same general idea: a multivariate object is expressed through constant, univariate, bivariate, and higher-order terms.

Like HDMR, EMPR organizes information hierarchically by interaction order. Its main distinction is the use of support functions, which define how lower-order terms are extended across the remaining variables.

This makes EMPR a support-based representation of multivariate structure rather than only a purely additive decomposition.

Terms and Components

Both HDMR and EMPR are organized by term order.

Zeroth-order term

The zeroth-order term f0 is the constant baseline. It captures the global reference level of the function or tensor.

First-order terms

A first-order term depends on a single variable, for example fi(xi). It describes the isolated contribution of one variable, independent of interactions with other variables.

Second-order terms

A second-order term depends on two variables, for example fij(xi, xj). It captures pairwise interactions that cannot be explained by the corresponding first-order terms alone.

Higher-order terms

Higher-order terms involve three or more variables, such as fijk(xi, xj, xk). These terms represent increasingly complex interactions.

Truncation

A decomposition truncated at order m keeps only terms up to that order:

\[f^{(m)}(\mathbf{x}) = \sum_{\lvert u \rvert \le m} f_u(\mathbf{x}_u)\]

This is the mathematical basis for lower-order approximation and reconstruction.

Support Vectors and Weights

In HDMRLib, both HDMR and EMPR are implemented for tensor-valued data using per-dimension support vectors.

Let the tensor have d dimensions. Then each dimension is associated with a support vector si ∈ ℝni, where ni is the size of the i-th mode.

HDMR in the library

In the current HDMR implementation, the constant term is obtained by successive contractions of the tensor with weighted support vectors:

\[g_0 = G \times_1 (s_1 \odot w_1)^\top \times_2 (s_2 \odot w_2)^\top \cdots \times_d (s_d \odot w_d)^\top\]

Here:

  • G is the input tensor

  • si is the support vector for dimension i

  • wi is the corresponding weight vector

  • ⊙ denotes elementwise multiplication

  • ×k denotes contraction along mode k

Higher-order HDMR terms are then obtained recursively after removing lower-order contributions.

EMPR in the library

In the current EMPR implementation, the constant term is also obtained by successive contractions, but the construction uses support vectors together with per-dimension scalar normalization factors:

\[g_0 = \left( \cdots \left( G \times_1 s_1^\top \right)\alpha_1 \times_2 s_2^\top \right)\alpha_2 \cdots \times_d s_d^\top \alpha_d\]

In the current implementation,

\[\alpha_i = \frac{1}{n_i}\]

for a mode of size ni.

This reflects the current code path in HDMRLib: EMPR uses support vectors directly, while HDMR uses support vectors together with explicit weight vectors.

HDMR and EMPR in Tensor Form

For tensor-valued data, the decomposition is not interpreted only as a function expansion, but also as a structured representation of a multiway array.

In the library implementation, a component term defined on a subset of dimensions is expanded back to the full tensor shape by combining it with support vectors along the remaining dimensions. This gives a tensor-level analogue of lower-order functional decomposition.

As a result, both methods support the same high-level ideas:

  • decompose a high-dimensional object into lower-order terms

  • retain only terms up to a chosen order

  • reconstruct an approximation from the retained terms

HDMR vs EMPR

HDMR and EMPR are closely related, but they are not identical.

Common structure

Both methods:

  • organize information hierarchically by order

  • represent a multivariate object through lower-order terms

  • support truncation to obtain lower-order approximations

Main difference

The main conceptual difference is the role of support functions.

  • HDMR is most naturally introduced as a hierarchical functional decomposition

  • EMPR uses support functions explicitly as part of the representation

In the current HDMRLib implementation, both methods operate on tensors and both rely on support vectors at the implementation level. The difference appears in how the component terms are computed:

  • HDMR uses weighted support vectors

  • EMPR uses support vectors together with per-dimension scalar normalization

References

  • H. Rabitz and Ö. F. Aliş, General foundations of high-dimensional model representations, Journal of Mathematical Chemistry, 1999.

  • B. Tunga and M. Demiralp, The influence of the support functions on the quality of enhanced multivariance product representation, Journal of Mathematical Chemistry, 2010.