Linear-in-Sums Models

• Linear-in-Sums Models are mathematical constructions that encode sums with a linear structure, integrating techniques from combinatorics, algebra, and statistics.
• They reveal deep structural properties via geometric valuations, recurrence identities, and operator-theoretic frameworks that bridge discrete and continuous methods.
• Their computational strategies, including the Alt summation formula, enable efficient evaluation and facilitate the unification of algebraic and statistical constraints.

Linear-in-Sums Models are a class of mathematical and statistical constructions where key quantities—such as rational functions, series, or expressions in algebraic structures—are encoded as sums that are linear either in their terms, indices, or combinatorial parameters. These models appear throughout algebraic combinatorics, analysis, commutative algebra, coding theory, and statistics. Linear-in-sums formulations often reveal underlying structural properties when sums are “packaged” into geometric, algebraic, or operator-theoretic frameworks. Notably, they unify disparate techniques, including valuations on cones, decompositions of ideals, recurrence relations for sequences, and combinatorial sum identities.

1. Geometric and Algebraic Foundations via Valuations on Cones

In combinatorial settings, linear-in-sums models arise when sums over posets—specifically, sums over linear extensions—are reconstructed as valuations on polyhedral cones. For a finite poset $P$ on $\{1,\dots,n\}$, two cones are associated:

• The root cone $$\mathrm{root}_P = \mathrm{pos}\{\mathbf{e}_i - \mathbf{e}_j : i <_P j\}$$, which lives in the hyperplane $\sum x_i = 0$.
• The weight cone $$\mathrm{wt}_P = \{\mathbf{x} \in \mathbb{R}^n_+ : x_i \geq x_j \text{ for } i <_P j\}$$.

A valuation on a cone $K$ is defined by the multivariate Laplace transform,

$$s(K; \mathbf{x}) = \int_K e^{-\langle \mathbf{x}, \mathbf{v} \rangle} d\mathbf{v},$$

which packages the sum over linear extensions into a single integral. For example,

$$\Psi_P(\mathbf{x}) = s(\mathrm{root}_P; \mathbf{x}), \qquad \Phi_P(\mathbf{x}) = s(\mathrm{wt}_P; \mathbf{x}),$$

where each is a rational function whose denominator is linear in sums (in the entries of $\mathbf{x}$, e.g. $(x_i-x_j)$ or partial sums). In cases such as strongly planar posets or forests, these factorizations yield product formulas (e.g., Greene's theorem) where the combinatorial structure of $P$ is reflected as circuits or generators in the associated cone or semigroup ring.

From an algebraic perspective, the Hilbert series of the affine semigroup ring of $K$,

$$H(K; \mathbf{X}) = \sum_{\mathbf{v} \in K \cap L} \mathbf{X}^{\mathbf{v}} \qquad \text{with}\quad \mathbf{X}^{\mathbf{v}} = e^{\langle \mathbf{x},\mathbf{v}\rangle},$$

encodes valuations and so connects linear-sums identities to the structure of toric ideals and complete intersections. The lowest degree homogeneous term recovers $s(K; \mathbf{x})$ up to sign.

2. Statistical and Statistical Learning Formulations

Linear-in-sums models are foundational in regression, estimation, and covariance decompositions where the objective is often formulated as a sum of quadratic forms or linear combinations subject to constraints. For instance, in sparse unit-sum regression models (Koning et al., 2019), the solution to

$$\min_{\beta \in \mathbb{R}^m} \| y - X\beta \|_2^2 \quad \text{subject to}\quad \sum_{i=1}^m \beta_i = 1,\quad \|\beta\|_0 \leq k,\quad \|\beta\|_1 \leq 1 + 2s$$

balances linear-in-sum constraints (unit sum, sparsity, and $\ell_1$ norm) to construct sparse portfolios or estimators. The underlying linear structure enables explicit trade-offs between sparsity and shrinkage and is key to predictive performance.

Covariance decomposition models (Frisch scheme (Ning et al., 2013)) also fit in the linear-in-sums paradigm by decomposing $\Sigma$ as

$\Sigma = \hat{\Sigma} + D,$

where $D$ is diagonal, and finding $\hat{\Sigma}$ of minimal rank. The rank minimization is relaxed to trace minimization, and the estimator structure is governed by linear sum constraints.

3. Componentwise Linear Ideals and Algebraic Sums

In commutative algebra, linear-in-sums behavior governs when the sum of componentwise linear ideals preserves componentwise linearity (Dao et al., 7 Apr 2025). Necessary and sufficient conditions are established, especially for $S=k[x,y]$, where the sum $I+J$ is componentwise linear if either the orders or regularities of intersections $I\cap J$ align with those of $I$ and $J$, and further compatibility conditions are satisfied. The central construction in higher dimensions relies on assembling ideals as

$I = \sum_{f \in \mathcal{L}} f \cdot I_f$

with $\mathcal{L}$ a collection of squarefree monomials and $I_f$ monomial ideals, provided a compatibility condition holds among ideals indexed by $\mathcal{L}$. This encodes the sum operation as linear in the summands and propagates linear resolutions structurally.

4. Linear Recurrence Sums and Bell Polynomial Formulation

Linear-in-sums models are apparent in recurrence sequences and their sums (Birmajer et al., 2015). For any homogeneous linear recurrence $\{a_n\}$ of order $d$, arithmetic subsequences $a_{mn + r}$ satisfy their own linear recurrence, with explicit coefficients given via partial Bell polynomials evaluated at a generalized Lucas sequence. Cumulative sums

$S_n = \sum_{k=0}^n a_k$

admit elegant closed-form expressions such as

$$q(1) \sum_{j=0}^n a_j = \sum_{j=0}^{d-1} \left(\sum_{i=0}^{d-1-j} c_i\right) (a_{n+j+1} - a_j),$$

with $q(1) = 1 - c_1 - \cdots - c_d$. These identities underpin models where linear summation is structurally preserved—particularly for estimation and forecasting in signal processing and time series.

5. Linear Sum Identities in Analytic Number Theory and Combinatorics

In analytic number theory, linear Euler sums and their decompositions into Tornheim double series (Adegoke, 2015), as well as explicit formulas for alternating Euler T-sums (Wang et al., 2020), exemplify linear-in-sums structure. The results demonstrate that sums such as

$E(n, r) = \sum_{p=1}^n T(r-1, n-p+1, p)$

can be written as finite linear combinations of double series, and, strikingly, certain nontrivial linear combinations simplify entirely to Riemann zeta values.

Ohno sums of multiple zeta values (Hirose et al., 2019) and their $\mathbb{Q}$-linear relations further extend this theme, establishing new families of relations that are not implied by existing duality (Ohno's relation) and uncovering algebraic symmetries through generating series.

6. Operator-Theoretic and Topological Formulations

Linear-in-sums models also manifest in topological and operator-theoretic contexts (Dubrovin, 2019). For modules of formal series or maps vanishing on large sets determined by a filter, infinite sums (formal linear sums) are made rigorous via a topology induced by the filter. Operators on such spaces are characterized by matrices whose rows belong to spaces of $\mathcal{S}$-zero maps, and continuity is equivalent to a linear-in-sums condition on the matrix entries and their zero sets. This brings linear-in-sums models into the field of infinite-dimensional algebra and module theory.

7. Computational Summation Techniques

The efficient approximation of sums in linear-in-sums models is advanced by the Alt summation formula (Pinelis, 2017), which uses linear combinations of integrals rather than derivatives to approximate sums: $$\sum_{k=0}^{n-1} f(k) \approx \sum_{j=1-m}^{m-1} \tau_{m, 1+|j|} \int_{j/2 - 1/2}^{n - 1/2 - j/2} f(x)\, dx.$$ This approach extends to multi-index sums (over lattice polytopes), is computationally efficient, and is exact for polynomials up to degree $2m-1$. Such techniques are particularly well suited to numerical evaluation and analysis within large-scale linear-in-sums models.

Conclusion

Linear-in-Sums Models unify a broad spectrum of mathematical constructions—algebraic, geometric, analytic, and statistical—where summation and linearity intersect, often encoding deep structural results and efficient computation. Their formulation in terms of valuations, operator theory, ideal compositions, and combinatorial series allows researchers to pass between explicit summation identities and higher-level geometric or algebraic representations. The continued development of these models enables systematic generalizations, computational advances, and a refined understanding of the foundational structures in pure and applied mathematics.