The nuclear norm, also known as the trace norm, is a crucial concept in various fields, including machine learning, signal processing, and optimization. It plays a vital role in matrix completion and low-rank approximation problems. Understanding its convexity is essential for designing efficient algorithms and analyzing their performance. This article aims to provide a clear and comprehensive proof of the convexity of the nuclear norm.
Understanding the Nuclear Norm
Before delving into the proof, let's establish a solid understanding of the nuclear norm. For a given matrix A, its nuclear norm, denoted as ||A||<sub>*</sub>, is defined as the sum of its singular values.
Definition: The nuclear norm of a matrix A is defined as:
||A||<sub>*</sub> = Σ<sub>i</sub> σ<sub>i</sub>(A)
where σ<sub>i</sub>(A) represents the i-th singular value of A.
Singular values are non-negative and represent the magnitudes of the principal axes of the ellipsoid formed by the image of the unit sphere under the linear transformation defined by A.
Convexity: A Fundamental Property
Convexity is a fundamental property in optimization. A function is considered convex if the line segment connecting any two points on its graph lies entirely above or on the graph itself. In mathematical terms, a function f(x) is convex if:
f(λx + (1-λ)y) ≤ λf(x) + (1-λ)f(y)
for all x, y in the domain of f and for all λ ∈ [0, 1].
Proving the Convexity of the Nuclear Norm
To prove the convexity of the nuclear norm, we will leverage the concept of the singular value decomposition (SVD). The SVD allows us to decompose any matrix A into three matrices:
A = UΣV<sup>T</sup>
where:
- U is an orthogonal matrix whose columns are the left singular vectors of A.
- Σ is a diagonal matrix with the singular values of A on the diagonal.
- V is an orthogonal matrix whose columns are the right singular vectors of A.
Using the SVD, we can express the nuclear norm of A as:
||A||<sub></sub> = ||Σ||<sub></sub>
Since Σ is a diagonal matrix, its nuclear norm is simply the sum of its diagonal elements, which are the singular values of A.
Now, consider two matrices A and B and a scalar λ ∈ [0, 1]. We want to prove that:
||λA + (1-λ)B||<sub></sub> ≤ λ||A||<sub></sub> + (1-λ)||B||<sub>*</sub>
Let A = U<sub>A</sub>Σ<sub>A</sub>V<sub>A</sub><sup>T</sup> and B = U<sub>B</sub>Σ<sub>B</sub>V<sub>B</sub><sup>T</sup> be the SVDs of A and B, respectively. Then:
λA + (1-λ)B = λU<sub>A</sub>Σ<sub>A</sub>V<sub>A</sub><sup>T</sup> + (1-λ)U<sub>B</sub>Σ<sub>B</sub>V<sub>B</sub><sup>T</sup>
This expression can be rewritten using the properties of matrix multiplication:
λA + (1-λ)B = U<sub>C</sub>(λΣ<sub>A</sub> + (1-λ)Σ<sub>B</sub>)V<sub>C</sub><sup>T</sup>
where U<sub>C</sub> and V<sub>C</sub> are orthogonal matrices obtained by appropriate combinations of U<sub>A</sub>, V<sub>A</sub>, U<sub>B</sub>, and V<sub>B</sub>.
Since the nuclear norm is invariant under orthogonal transformations (i.e., ||UAV<sup>T</sup>||<sub></sub> = ||A||<sub></sub> for orthogonal U and V), we can write:
||λA + (1-λ)B||<sub></sub> = ||λΣ<sub>A</sub> + (1-λ)Σ<sub>B</sub>||<sub></sub>
Now, we can leverage the fact that the sum of diagonal matrices is also a diagonal matrix. The singular values of the sum are simply the sum of the corresponding singular values of the individual matrices. Therefore:
||λΣ<sub>A</sub> + (1-λ)Σ<sub>B</sub>||<sub>*</sub> = Σ<sub>i</sub> |λσ<sub>i</sub>(A) + (1-λ)σ<sub>i</sub>(B)|
Using the triangle inequality (|a + b| ≤ |a| + |b|), we obtain:
Σ<sub>i</sub> |λσ<sub>i</sub>(A) + (1-λ)σ<sub>i</sub>(B)| ≤ Σ<sub>i</sub> (λ|σ<sub>i</sub>(A)| + (1-λ)|σ<sub>i</sub>(B)|)
This simplifies to:
||λA + (1-λ)B||<sub>*</sub> ≤ λΣ<sub>i</sub> |σ<sub>i</sub>(A)| + (1-λ)Σ<sub>i</sub> |σ<sub>i</sub>(B)|
Recognizing that the sum of singular values is the nuclear norm, we have:
||λA + (1-λ)B||<sub></sub> ≤ λ||A||<sub></sub> + (1-λ)||B||<sub>*</sub>
This inequality holds for any matrices A, B, and any scalar λ ∈ [0, 1], proving the convexity of the nuclear norm.
Conclusion
The convexity of the nuclear norm is a fundamental property with significant implications for various applications involving matrices. This proof, derived using the singular value decomposition and triangle inequality, establishes the convexity of the nuclear norm, providing a foundation for developing efficient optimization algorithms and analyzing their performance in areas like matrix completion, low-rank approximation, and compressed sensing. The understanding of the convexity of the nuclear norm plays a crucial role in advancing these fields and driving innovation in related areas.
