🔢 mathematics

Finiteness of Hadamard ranks

This paper classifies projective varieties for which the Hadamard rank is finite for any point, establishes the finiteness of this rank for various tensor varieties, and proves sharp upper bounds on the maximum Hadamard rank for specific families of algebraic varieties.

Original authors: Dario Antolini, Edoardo Ballico, Alessandro Oneto

Published 2026-04-01

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Dario Antolini, Edoardo Ballico, Alessandro Oneto

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 in a vast, multidimensional room filled with points. Each point is defined by a list of numbers (like coordinates on a map). Now, imagine you have a special rule for combining these points: instead of adding them together, you multiply their numbers one by one. This is called the Hadamard product.

The paper you shared is a mathematical detective story about a specific question: "If I pick any random point in this room, can I build it by multiplying together a few points from a specific shape (a 'variety'), and if so, how many points do I need?"

The number of points needed to build a target is called its Hadamard Rank. The big mystery the authors solve is: For which shapes is this number always finite? In other words, for which shapes can we always build any point in the room, no matter how weird it is?

Here is the breakdown of their discovery using simple analogies:

1. The Problem: The "Zero" Trap

Imagine your shape is a collection of Lego bricks. You want to build a specific tower (your target point) by snapping these bricks together.

The Good News: If your shape is "strongly concise" (a fancy math term meaning it doesn't get stuck in a corner where too many numbers are zero), you can build any point in the room.
The Bad News: If your shape is "lazy" and sits entirely inside a wall where one coordinate is always zero (like a flat sheet on the floor), you can never build a point that sticks up off that floor.
The Trap: Even if you aren't stuck in a wall, there's a sneaky trap. If your shape is "binomial" (defined by a specific type of equation like $x^2 - 3y^2 = 0$ ), it might be able to build most points, but there will be specific, weird points you can never reach, no matter how many bricks you use. The authors call these "infinite rank" points.

The Main Discovery (Theorem A):
The authors found a simple rule to tell if you can build everything. Your shape must be Strongly Concise.

Analogy: Think of a shape as a team of workers. To be "strongly concise," the team must be versatile enough that for every single task (coordinate), there is at least one worker who can do it without needing help from a specific "zero" condition. If the team is too rigid, they can't build the whole room.

2. The Application: Tensors and Data

Why does this matter? The paper mentions Tensors and Restricted Boltzmann Machines (used in AI and statistics).

The Analogy: Imagine you have a giant, complex spreadsheet of data (a tensor). You want to simplify it by breaking it down into smaller, simpler spreadsheets multiplied together.
The Result: The authors prove that for many important types of data structures (like Grassmannians, which describe geometric shapes, or Chow varieties, which describe how things break apart), you can always break down any complex data into a finite number of simple pieces. You don't have to worry about hitting a "dead end" where the data becomes impossible to decompose.

3. The Upper Limit: How Many Bricks Do I Need?

Once we know we can build any point, the next question is: "What is the maximum number of bricks I might ever need?"

The Old Way: Previously, mathematicians knew you could build points with non-zero numbers using a certain amount of effort. But points with zeros were tricky.
The New Bound (Theorem B): The authors prove that if your shape is "strongly concise" and doesn't have points with too many zeros, you will never need more bricks than the number of dimensions in the room.
- Analogy: If you are in a 3D room (length, width, height), you will never need more than 3 bricks to build any point, provided your bricks are "strongly concise." If you are in a 100-dimensional room, you'll never need more than 100.

4. The "Border" Case: The Illusion of Rank

There is a tricky concept called Border Rank.

The Analogy: Imagine you are trying to build a tower. You can get infinitely close to the target tower using 2 bricks, but you can never quite snap them together to make the exact target. You need a 3rd brick to actually finish it.
The Discovery: The authors show that for "nice" shapes (curves that don't have points with two zeros), the number of bricks you need to actually build the point is the same as the number you need to get close to it. The "illusion" of being able to do it with fewer bricks doesn't happen here.

Summary in One Sentence

The authors proved that as long as your mathematical "shape" is versatile enough to avoid getting stuck in zero-coordinate corners, you can build any point in the universe by multiplying together a finite (and surprisingly small) number of points from that shape.

Why is this cool?
It guarantees that in the world of data science and geometry, we don't have to fear "impossible" decompositions. If the shape is right, the math always works out, and we know exactly how much "effort" (number of components) is required to reconstruct the data.

1. Problem Statement

The paper addresses the finiteness of the Hadamard rank for points with respect to a projective variety $X \subset \mathbb{P}^N$ .

Context: The Hadamard product of two points $p, q \in \mathbb{P}^N$ is defined coordinate-wise as $p \star q = (p_0q_0 : \dots : p_Nq_N)$ . The Hadamard- $X$ -rank of a point $q$ , denoted $Hrk_X(q)$ , is the minimum number of points $p_1, \dots, p_m \in X$ such that $q = p_1 \star \dots \star p_m$ .
The Core Question: For which irreducible projective varieties $X$ is the Hadamard rank finite for every point in the ambient space $\mathbb{P}^N$ ?
Background: Previous work (e.g., [AMO25]) established conditions for the generic Hadamard rank to be finite (requiring $X$ to be "concise" and not contained in a binomial hypersurface). However, generic finiteness does not imply finiteness for all points, particularly those with zero coordinates. The paper seeks to classify varieties where the rank is finite universally, not just generically.

2. Methodology

The authors employ techniques from algebraic geometry, specifically focusing on:

Hadamard Products of Varieties: Iterative Hadamard powers $X^{\star m}$ and their Zariski closures.
Coordinate Hyperplanes: Analyzing the intersection of varieties with coordinate hyperplanes $H_i = \{x_i = 0\}$ and their unions.
Semi-continuity: Investigating the lower semi-continuity of the Hadamard rank function, particularly for curves.
Dimensional Arguments: Using Bertini's theorem and dimension counting to relate the rank of a general variety to that of a generic curve section.
Toric and Binomial Geometry: Utilizing properties of toric varieties and binomial hypersurfaces to identify obstructions to finiteness.

3. Key Contributions and Results

A. Classification of Varieties with Finite Maximum Rank (Theorem A)

The paper provides a complete classification of irreducible varieties for which the Hadamard rank is finite for every point in $\mathbb{P}^N$ .

Definition: A variety $X$ $X$ is called strongly concise if for every coordinate hyperplane $H_i$ $H_{i}$ , the intersection $X \cap H_i$ $X \cap H_{i}$ is not contained in the union of all other coordinate hyperplanes $\bigcup_{j \neq i} H_j$ $⋃_{j \neq = i} H_{j}$ .
- Algebraic Interpretation: The vanishing ideal of $X$ contains no polynomial of the form $x_i F + x^\alpha$ , where $x^\alpha$ is a monomial not involving $x_i$ .
Theorem A: The Hadamard- $X$ -rank is finite for every point $p \in \mathbb{P}^N$ if and only if $X$ is strongly concise.
Implication: This resolves the gap between generic finiteness and universal finiteness. If $X$ is not strongly concise, there exist points with infinite Hadamard rank (typically points with specific zero-coordinate patterns).

B. Upper Bounds on Maximum Rank (Theorem B)

Once finiteness is established, the authors derive sharp upper bounds on the maximum Hadamard rank ( $Hrk_{max}^X$ ).

Theorem B: Let $X \subset \mathbb{P}^N$ $X \subset P^{N}$ be an irreducible variety of dimension $n \leq N/2$ $n \leq N /2$ such that $X$ $X$ does not contain any point with at least $n+1$ $n + 1$ zero coordinates (i.e., $X \cap \Delta_{N-n-1} = \emptyset$ $X \cap Δ_{N - n - 1} = \emptyset$ , where $\Delta_k$ $Δ_{k}$ is the variety of points with at most $k+1$ $k + 1$ non-zero coordinates).
- Then, $Hrk_X(p) \leq N$ for any $p \in \mathbb{P}^N$ .
Sharpness: Example 4.9 demonstrates that this bound is sharp; for certain curves, the rank of a coordinate point is exactly $N$ .
Mechanism: The proof reduces the problem to the case of curves ( $n=1$ ) using a generic linear section (Bertini's Theorem). For curves avoiding points with two zero coordinates, the rank is shown to be lower semi-continuous and bounded by $N$ .

C. Applications to Tensor Decompositions

The authors apply Theorem A to several important families of varieties relevant to tensor decompositions, proving that their Hadamard ranks are universally finite:

Grassmannians: $G(k, n)$ in their Plücker embedding.
Tangential Varieties: To Veronese embeddings ( $\tau_{d,n}$ ).
Chow Varieties: Varieties of products of linear forms.
Varieties of Reducible Forms: Products of homogeneous polynomials.

Result: For any of these varieties (and their secant varieties), any tensor in the ambient space can be expressed as a coordinate-wise product of a finite number of tensors from the variety. This complements previous results limited to Segre-Veronese varieties.

D. Border Hadamard Rank and Semi-continuity

The paper investigates the relationship between the standard Hadamard rank and the Border Hadamard Rank (defined via the closure of Hadamard powers).

Pathology: The authors provide an example (Example 4.1) of a conic where the Hadamard rank is not lower semi-continuous. A point may lie in the closure of the set of rank-2 points (border rank 2) but require rank 3 to be decomposed exactly.
Condition for Equality: Lemma 4.5 and Theorem 4.6 establish that if a variety (specifically a curve) does not contain points with "too many" zero coordinates (specifically $X \cap \Delta_{N-n-1} = \emptyset$ ), then the rank equals the border rank in the low-rank regime, and the rank function is lower semi-continuous.

4. Significance

Theoretical Advancement: The paper moves beyond the study of generic properties in algebraic geometry to establish global properties of Hadamard ranks. It identifies the precise geometric obstruction (lack of strong conciseness) that causes infinite rank for specific points.
Tensor Decompositions: By proving the finiteness of Hadamard ranks for Grassmannians, Chow varieties, and reducible forms, the paper validates the use of Hadamard decompositions for a much broader class of tensors than previously known. This has potential implications for statistical models (like Restricted Boltzmann Machines) and signal processing where such decompositions are used.
Sharp Bounds: The establishment of the bound $N$ for varieties with "sparse" zero coordinates provides a concrete limit for algorithmic complexity in finding such decompositions.
New Concepts: The formalization of "strongly concise" varieties and the analysis of the gap between rank and border rank in the multiplicative (Hadamard) setting enrich the toolkit of algebraic geometers working with tensor decompositions.

In summary, this paper provides a definitive classification of when Hadamard decompositions are universally possible and quantifies the complexity (rank) of such decompositions for major classes of algebraic varieties.