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Euclidean distance degree defect of singular projective varieties

This paper extends previous results on the Euclidean distance degree defect from smooth to arbitrary complex projective varieties by providing a constructible enhancement and a topological formula to compute the gap between the unit and generic ED degrees, thereby offering a new method for determining ED degrees in broad generality.

Original authors: Laurenţiu G. Maxim, Jose Israel Rodriguez, Botong Wang

Published 2026-05-14

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Laurenţiu G. Maxim, Jose Israel Rodriguez, Botong Wang

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are standing in a vast, multi-dimensional landscape (mathematicians call this a "projective variety"). You have a specific destination in mind: the closest point on this landscape to where you are currently standing. This is a common problem in data science and engineering, often called a "nearest-point problem."

In the world of pure math, there are two ways to measure how "complicated" finding this closest point is:

The "Generic" Way (The Smooth Path): Imagine the landscape is perfectly smooth and you are approaching it from a random, average direction. The number of possible "critical points" (places where the math gets tricky or stops) you might encounter is called the Generic Euclidean Distance (ED) Degree. Think of this as the standard, easy-to-calculate difficulty rating for a smooth hill.
The "Unit" Way (The Real-World Path): Now, imagine the landscape has bumps, cracks, or sharp corners (singularities), and you are approaching it with a specific, standard set of rules (unit weights). The number of critical points here is the Unit ED Degree. This is often harder to calculate because the "bumps" in the landscape can confuse the math.

The Problem: The "Defect"

Sometimes, the "Unit" difficulty is much lower than the "Generic" difficulty. The paper calls this difference the ED Degree Defect.

Think of it like this:

Generic ED Degree: The number of ways you could get stuck if you were driving a car on a perfectly paved, straight road with no traffic. (High number of theoretical possibilities).
Unit ED Degree: The number of ways you actually get stuck when driving that same road, but it's raining and you have a specific type of car. (Fewer possibilities because the conditions limit your options).
The Defect: The gap between those two numbers.

The authors noticed that in many real-world models (like analyzing data or optimizing engineering designs), this gap is huge. However, calculating the "Unit" number directly is often a nightmare, while the "Generic" number is easier to find.

The Solution: A Topological Map

The paper provides a new "map" to calculate that gap (the defect) without having to solve the impossible "Unit" problem directly.

Instead of trying to drive the car through the storm, the authors say: "Look at the shape of the road's cracks."

They use a concept from topology (the study of shapes) called Vanishing Cycles. Imagine the landscape is a rubber sheet. When you stretch it or poke it (representing the mathematical "singularities" or cracks), the way the rubber snaps back or creates a hole tells you something about the complexity.

The paper's main formula says:

The Defect = (A specific count of the "holes" and "twists" in the cracks of the landscape) × (The Generic Difficulty of those cracks).

In simpler terms, they found a way to measure the "roughness" of the singular points (the cracks) and use that measurement to predict exactly how much the "Unit" difficulty drops compared to the "Generic" difficulty.

How They Did It

The Pencil of Quadrics: They imagined a family of shapes (like a pencil of pencils, but made of curved surfaces) that slowly morphs from a "smooth" shape to the "cracked" shape.
The Vanishing Cycle: As the shape changes, they tracked how the "holes" in the shape appear or disappear.
The Formula: They proved that the difference between the two difficulty scores is exactly equal to the sum of the "Euler characteristics" (a topological count of holes and twists) of these vanishing cycles, weighted by how complex the singularities are.

Why This Matters (According to the Paper)

The authors show that for many complex models used in statistics and engineering (like analyzing low-rank matrices or specific data structures), the "Unit" difficulty is often much lower than the "Generic" one.

Previously, if a model had a crack (singularity), mathematicians couldn't easily calculate the true difficulty (Unit ED Degree). They had to rely on the "Generic" number, which was an overestimate.

This paper gives them a tool to:

Look at the cracks (singularities).
Measure their topological "roughness."
Calculate the exact "Defect."
Subtract that defect from the easy-to-find "Generic" number to get the true, accurate "Unit" number.

Real-World Examples in the Paper

The authors tested their new "map" on specific shapes:

A Surface with a Pinch: They looked at a 3D shape that had a single sharp point. They calculated the "roughness" of that point and found the defect was exactly 1.
The "Whitney Umbrella": A shape that looks like an umbrella with a handle that folds into itself. They calculated the defect for this shape and found it was also 1.
Sparse Matrices: They looked at grids of numbers where many spots are forced to be zero (like a puzzle with missing pieces). They found that even with these simple constraints, the "defect" can be large (e.g., a gap of 28), meaning the generic calculation would be wildly inaccurate without their new formula.

In summary: The paper provides a mathematical "translation key" that allows researchers to understand how the "cracks" in a complex shape reduce the difficulty of finding the nearest point. Instead of struggling to solve the hard problem directly, they can now measure the cracks and use a topological formula to find the answer.

Technical Summary: Euclidean Distance Degree Defect of Singular Projective Varieties

Problem Statement
The Euclidean distance degree (ED degree) is a fundamental invariant in algebraic geometry and applied mathematics, measuring the algebraic complexity of nearest-point problems on an algebraic variety. Specifically, it counts the number of complex critical points of the squared Euclidean distance function from a generic data point to the variety. Two primary variants are studied: the unit ED degree ($uEDdeg$), corresponding to the standard Euclidean metric, and the generic ED degree ($gEDdeg$), corresponding to a generic weighted metric.

While $gEDdeg$ is often more tractable to compute (as the associated quadric intersects the variety transversally), $uEDdeg$ is frequently more difficult to determine due to non-transversal intersections with the isotropic quadric. In many applications spanning optimization, engineering, statistics, and data science, a significant gap exists between these two values. This paper defines the ED degree defect as the difference:
$\text{EDdefect}(X) := gEDdeg(X) - uEDdeg(X)$
The central problem addressed is the computation of this defect for arbitrary complex projective varieties, particularly those that are singular. Previous results, such as those in [15], were limited to the smooth setting. Since many models in applications are singular, a generalization of the topological formulas for the defect to the singular case is required.

Methodology
The authors extend the framework of pencils of quadric hypersurfaces and vanishing cycles to singular varieties. The methodology relies on the following constructs:

Pencil of Quadrics: For a variety $X \subset \mathbb{P}^n$ and a generic weight $w$ , the authors consider a pencil of quadrics $Q_\lambda$ interpolating between the isotropic quadric $Q$ (associated with the unit metric) and the weighted quadric $Q_w$ (associated with the generic metric). This induces a pencil of hypersurfaces $X_\lambda = X \cap Q_\lambda$ on $X$ .
Incidence Variety and Function: They define an incidence variety $\tilde{X}$ and a regular function $f = f_0/f_\infty$ on $X \setminus X_\infty$ , where $f_0$ and $f_\infty$ define the quadrics $Q$ and $Q_w$ respectively.
Vanishing Cycles: The core of the approach utilizes the vanishing cycle functor $\phi_f$ applied to the constructible function of the local Euler obstruction, $Eu_X$ . This functor captures the topological change in the fibers of $f$ near the singular locus of the pencil.
Stratification: To handle singularities, the authors employ Whitney stratifications. They analyze the behavior of the constructible function $\phi_f(Eu_{X \setminus Q_w})$ restricted to the singular locus of the function $f$ (which corresponds to the intersection of $X$ with the isotropic quadric).
Basis Expansion: The defect is expressed as a linear combination of generic ED degrees of the strata in the singular locus, weighted by coefficients derived from the topology of complex links between strata.

Key Contributions and Results

Topological Formula for the Defect (Theorem 1.6): The paper provides a constructible enhancement and a topological formula for the ED degree defect of an arbitrary complex projective variety $X$ . The defect is given by the Euler characteristic of the vanishing cycle of the local Euler obstruction function, restricted to the intersection of the variety with the isotropic quadric (minus a generic hyperplane):
$\text{EDdefect}(X) = (-1)^{\dim(X)-1} \chi \left( (\phi_f(Eu_{X \setminus Q_w}))|_{X_0 \setminus (Q_w \cup H)} \right)$
This generalizes previous results from [15] which were restricted to smooth varieties.
Stratified Formula (Theorem 1.7): For varieties with singularities, the authors derive a formula expressing the defect as a sum over the strata of the singular locus $Z = X_0 \cap \text{Sing}(f)$ . The defect is a weighted sum of the generic ED degrees of the strata:
$\text{EDdefect}(X) = \sum_{V \in \mathcal{X}_0} (-1)^{\text{codim}_{\mathcal{X}_0}(V)} \alpha_V \cdot gEDdeg(V)$
The coefficients $\alpha_V$ are determined by the values of the constructible function $\mu$ and the topology of complex links between strata.
Isolated Singularities (Corollary 1.8): In the case where the singularities are isolated, the formula simplifies to a sum involving the values of the constructible function at the singular points, directly relating the defect to Milnor fiber data.
Computational Examples: The paper validates these theoretical results through concrete examples:
- Example 3.1 & 3.2: Calculations for specific singular surfaces in $\mathbb{P}^3$ with isolated singularities and Whitney umbrella singularities, respectively. The results ($EDdefect = 1$ in both cases) are verified against symbolic computations of the full ED degrees.
- Structured Low Rank Matrices (Section 4.1): The authors apply their framework to rank-constrained matrices with sparsity patterns (e.g., $X_{3,3,2} \cap V(x_0)$ ). They report specific defect values (e.g., 28) and highlight the computational challenges in determining the local Euler obstruction on the singular loci of these intersections.
- Toric Hypersurfaces (Section 4.2): The paper presents a table of computed defects for binomial hypersurfaces and their intersections with the sphere, noting that while the defect for the binomial itself is zero, the intersection with the sphere introduces non-trivial defects.

Significance and Claims
The paper claims to fill a critical void in the literature by providing the first topological formulas for the ED degree defect applicable to singular projective varieties. The authors state that their approach offers a new method for computing ED degrees in broad generality, leveraging the fact that the generic ED degree is typically more tractable than the unit ED degree.

The significance lies in enabling the analysis of models common in applied fields (such as structured low-rank approximation and toric models) that are inherently singular. By expressing the defect in terms of invariants of the singularities (Milnor fibers, complex links, and local Euler obstructions), the paper provides a theoretical bridge between singularity theory and optimization complexity. The authors note that these tools are necessary to address open problems, such as the duality conjecture in structured low-rank approximation and extending minimal algebraic complexity results to higher-rank matrices.

The paper does not claim to provide a closed-form combinatorial formula for all cases but rather a topological framework that reduces the computation of the defect to the computation of local invariants and generic ED degrees of lower-dimensional strata. It acknowledges that computing these invariants (particularly the local Euler obstruction on complex singular loci) remains a computational challenge, motivating further work in symbolic and numerical methods.