[Paper Review] Reciprocal maximum likelihood degrees of diagonal linear concentration models

Christopher Eur, Tara Fife+2 more|arXiv (Cornell University)|Nov 28, 2020

Stochastic processes and statistical mechanics13 references4 citations

TL;DR

This paper establishes a closed-form formula for the reciprocal maximum likelihood degree (rmld) of diagonal linear concentration models in terms of the characteristic polynomial of their associated matroid. It proves that rmld(ℒ) = (−2)^r ⋅ χ_M(1/2), where r is the dimension of the model and χ_M is the characteristic polynomial of the matroid M, resolving a conjecture by Sturmfels, Timme, and Zwiernik on the polynomiality of rmld for general diagonal models.

ABSTRACT

We show that the reciprocal maximal likelihood degree (rmld) of a diagonal linear concentration model $\mathcal L \subseteq \mathbb{C}^n$ of dimension $r$ is equal to $(-2)^rχ_M( extstyle\frac{1}{2})$, where $χ_M$ is the characteristic polynomial of the matroid $M$ associated to $\mathcal L$. In particular, this establishes the polynomiality of the rmld for general diagonal linear concentration models, positively answering a question of Sturmfels, Timme, and Zwiernik.

Motivation & Objective

• To resolve a conjecture by Sturmfels, Timme, and Zwiernik on the polynomiality of the reciprocal maximum likelihood degree (rmld) for general diagonal linear concentration models.
• To establish a precise algebraic formula linking rmld to the matroid-theoretic invariants of the model.
• To provide a generalization of the ML degree formula to higher-order systems involving d-th powers, extending known results for d=1.
• To clarify the geometric and combinatorial structure underlying critical points of the log-likelihood function on reciprocal varieties.

Proposed method

• Derives an alternate characterization of rmld using a system of equations in (ℂ*)^n: (x₁⁻¹,…,xₙ⁻¹) ∈ ℒ and (s₁x₁²−x₁,…,sₙxₙ²−xₙ) ∈ ℒ⊥ for generic sᵢ.
• Introduces a generalized counting function 𝒟(ℒ,d) for solutions to a d-th power system, showing 𝒟(ℒ,d) = (−d)^r ⋅ χ_M(1/d).
• Uses the Möbius function and deletion-contraction relations on the lattice of flats of the matroid M to derive the formula via inclusion-exclusion and duality.
• Applies the Tutte polynomial identity d^r ⋅ T_M(1−1/d, 0) = (−d)^r ⋅ χ_M(1/d) to connect to known matroid invariants.
• Employs Poincaré duality and cohomological tools to relate the solution count to the Poincaré polynomial of the complement of the hyperplane arrangement.
• Validates the formula through explicit computation on uniform matroids, recovering known polynomial expressions for rmld in the general case.

Experimental results

Research questions

• RQ1 Is the reciprocal maximum likelihood degree of a general diagonal linear concentration model a polynomial in n of degree r−1, as conjectured?
• RQ2 Can the rmld be expressed in terms of matroid invariants such as the characteristic polynomial or Tutte polynomial?
• RQ3 What is the geometric and algebraic mechanism behind the critical point count on the reciprocal variety of a linear concentration model?
• RQ4 How does the rmld behave under matroid operations like deletion and contraction, and can this be reflected in the solution count?
• RQ5 Does the formula 𝒟(ℒ,d) = (−d)^r ⋅ χ_M(1/d) generalize the known case d=1 and unify non-reciprocal and reciprocal ML degrees?

Key findings

• The reciprocal maximum likelihood degree of a diagonal linear concentration model ℒ of dimension r is exactly (−2)^r ⋅ χ_M(1/2), where χ_M is the characteristic polynomial of the associated matroid M.
• This formula confirms that rmld(ℒ) is a polynomial in n of degree r−1 for general models, answering a question posed by Sturmfels, Timme, and Zwiernik.
• For a general r-dimensional model in ℂ^[n], rmld(ℒ) = ∑_{i=1}^r (n−i−1 choose r−i) ⋅ 2^{r−i}, with explicit examples: 2n²−8n+7 for r=3 and 4/3n³−10n²+68/3n−15 for r=4.
• The formula 𝒟(ℒ,d) = (−d)^r ⋅ χ_M(1/d) generalizes the rmld to higher-order systems, with d=2 recovering the reciprocal case.
• The rmld is always odd unless zero, a pattern observed in numerical data and now explained by the formula.
• The deletion-contraction recurrence for rmld is fully characterized: it vanishes on loops, scales by (d−1) on coloops, and satisfies a linear recurrence otherwise.