[Paper Review] The Euclidean distance degree of an algebraic variety

Jan Draisma, Emil Horobeţ+3 more|arXiv (Cornell University)|Aug 31, 2013

Polynomial and algebraic computation23 references20 citations

TL;DR

This paper introduces the Euclidean Distance Degree (ED degree) as a fundamental invariant in algebraic geometry and optimization, measuring the number of critical points of the squared distance function from a general point to a real algebraic variety. It develops computational tools using Lagrange multipliers and algebraic geometry to compute the ED degree and average ED degree, with applications to low-rank matrices, tensors, and geometric models, proving duality invariance and deriving explicit formulas for key cases such as Segre varieties and rank-one tensors.

ABSTRACT

The nearest point map of a real algebraic variety with respect to Euclidean distance is an algebraic function. For instance, for varieties of low rank matrices, the Eckart-Young Theorem states that this map is given by the singular value decomposition. This article develops a theory of such nearest point maps from the perspective of computational algebraic geometry. The Euclidean distance degree of a variety is the number of critical points of the squared distance to a generic point outside the variety. Focusing on varieties seen in applications, we present numerous tools for exact computations.

Motivation & Objective

• To formalize the Euclidean Distance Degree (ED degree) as a measure of algebraic complexity in optimization over real algebraic varieties.
• To develop computational techniques using algebraic geometry and Lagrange multipliers for exact ED degree computation.
• To study the average ED degree (aED degree) under Gaussian sampling, linking it to random matrix theory and statistical models.
• To establish duality invariance: ED degree(X) = ED degree(X*) for projective varieties and relate it to polar classes.
• To derive explicit formulas for ED degree in key applications, including low-rank matrix varieties and Segre embeddings of tensors.

Proposed method

• Define the ED degree as the number of non-singular critical points of the squared Euclidean distance function from a general point to a variety.
• Use Lagrange multipliers to characterize critical points via the condition that the vector (u−x) is orthogonal to the tangent space TxX.
• Introduce the ED correspondence EX as the variety of pairs (x,u) where x is a critical point of du on X, enabling computation of average ED degree.
• Lift the ED correspondence to the conormal variety and use duality to prove ED degree(X) = ED degree(X*) for projective varieties.
• For rank-one tensors, derive a formula for the average ED degree as an expectation of the absolute determinant of a random symmetric matrix with Gaussian entries.
• Apply these tools to compute ED degrees for specific varieties such as the cardioid, low-rank matrices, and Segre varieties, using Macaulay2 code and symbolic computation.

Experimental results

Research questions

• RQ1 What is the algebraic complexity of finding the nearest point on a real algebraic variety to a given point in Euclidean space?
• RQ2 How can the ED degree be computed exactly for varieties arising in applications such as low-rank matrix completion and tensor decomposition?
• RQ3 What is the relationship between the ED degree of a variety and its dual variety, and does ED degree remain invariant under duality?
• RQ4 How does the average ED degree (aED degree) behave under Gaussian sampling, and can it be expressed via random matrix expectations?
• RQ5 Can closed-form or efficient formulas be derived for the ED degree of Segre varieties and other tensor models?

Key findings

• The ED degree of the cardioid is 3, with three real critical points for general external points, and the ED discriminant (evolute) separates regions with 3 or 1 real solutions.
• For the Segre variety of rank-one tensors in format $m_1 \times \cdots \times m_p$, the ED degree is given by Theorem 8.1 as $\sum_{i=1}^p \binom{m_i}{2} + 1$, with explicit values such as 6 for $2\times2\times2$ and 120 for $2^5$.
• The average ED degree for the $2\times2\times2$ tensor format is approximately 4.287, significantly less than its ED degree of 6, indicating that most critical points are complex.
• The average ED degree of a Segre variety under standard Gaussian measure equals $\frac{\pi^{p/2}}{2^{m/2} \prod_{i=1}^p \Gamma(m_i/2)} \cdot \mathbb{E}[|\det(A)|]$, where $A$ is a random symmetric matrix with independent Gaussian entries.
• For $p \geq 3$, the average ED degree stabilizes or slightly decreases beyond the bound $m_p - 1 \geq \sum_{i=1}^{p-1} (m_i - 1)$, though no geometric explanation is yet known.
• The duality property holds: $\mathrm{EDdegree}(X) = \mathrm{EDdegree}(X^*)$ for projective varieties, and the ED degree equals the sum of classical polar classes of $X$.