Do Semi-Definite Tensors Form A (convex) Cone?

Introduction

In the realm of convex analysis, a fundamental concept is the notion of a convex cone. A set $C$ is said to be a convex cone if it satisfies the following property: for any two elements $x$ and $y$ in $C$, and for any non-negative real numbers $\alpha$ and $\beta$, the linear combination $\alpha x + \beta y$ is also in $C$. This definition has far-reaching implications in various fields, including optimization, machine learning, and signal processing.

Convex Cones and Their Importance

Convex cones play a crucial role in convex analysis, as they provide a way to describe sets that are closed under linear combinations. This property makes them particularly useful in optimization problems, where the goal is to find the minimum or maximum of a function subject to certain constraints. In such problems, convex cones can be used to model the feasible region, which is the set of all possible solutions.

Semi-Definite Tensors

Semi-definite tensors are a generalization of semi-definite matrices to higher-order tensors. A tensor is said to be semi-definite if it can be written as the sum of the outer products of vectors, where each vector is a non-negative multiple of a standard basis vector. In other words, a tensor $T$ is semi-definite if there exists a set of vectors $v_1, v_2, \ldots, v_n$ and non-negative real numbers $\lambda_1, \lambda_2, \ldots, \lambda_n$ such that:

$$T = \sum_{i=1}^n \lambda_i v_i \otimes v_i$$

where $\otimes$ denotes the outer product.

Positive Semidefinite Tensors

A tensor $T$ is said to be positive semi-definite (PSD) if it is semi-definite and has no negative eigenvalues. In other words, a tensor $T$ is PSD if it can be written as the sum of the outer products of vectors, where each vector is a non-negative multiple of a standard basis vector, and all the eigenvalues of $T$ are non-negative.

The Convex Cone of Semi-Definite Tensors

The set of all semi-definite tensors forms a convex cone, denoted by $\mathcal{S}^n_+$. This cone is defined as the set of all tensors $T$ that can be written as the sum of the outer products of vectors, where each vector is a non-negative multiple of a standard basis vector.

Properties of the Convex Cone of Semi-Definite Tensors

The convex cone of semi-definite tensors has several important properties. First, it is closed under linear combinations, meaning that if $T_1$ and $T_2$ are both semi-definite tensors, then $\alpha T_1 + \beta T_2$ is also a semi-definite tensor for any non-negative real numbers $\alpha$ and $\beta$. Second, it is closed under scaling, meaning that if $T$ is a semi-definite tensor, then $\alpha T$ is also a semi-definite tensor for any non-negative real number $\alpha$.

Relationship with Other Convex Cones

The convex cone of semi-definite tensors is related to other convex cones in several ways. First, is a subset of the convex cone of positive semi-definite tensors, denoted by $\mathcal{S}^n_{++}$. Second, it is a subset of the convex cone of symmetric tensors, denoted by $\mathcal{S}^n$. Finally, it is a subset of the convex cone of non-negative tensors, denoted by $\mathcal{N}^n$.

Applications of Semi-Definite Tensors

Semi-definite tensors have several applications in various fields, including optimization, machine learning, and signal processing. In optimization, semi-definite tensors can be used to model the feasible region of a problem, which is the set of all possible solutions. In machine learning, semi-definite tensors can be used to model the covariance matrix of a multivariate distribution. In signal processing, semi-definite tensors can be used to model the autocorrelation matrix of a signal.

Conclusion

In conclusion, semi-definite tensors form a convex cone, denoted by $\mathcal{S}^n_+$. This cone is defined as the set of all tensors $T$ that can be written as the sum of the outer products of vectors, where each vector is a non-negative multiple of a standard basis vector. The convex cone of semi-definite tensors has several important properties, including closure under linear combinations and scaling. It is also related to other convex cones, including the convex cone of positive semi-definite tensors, the convex cone of symmetric tensors, and the convex cone of non-negative tensors. Finally, semi-definite tensors have several applications in various fields, including optimization, machine learning, and signal processing.

References

• [1] Horn, R. A., & Johnson, C. R. (1990). Matrix analysis. Cambridge University Press.
• [2] Lim, L. H. (2015). A tutorial on tensor analysis. Journal of Mathematical Imaging and Vision, 53(2), 147-164.
• [3] Qi, L. (2012). Eigenvalues of a real supersymmetric tensor. Journal of Mathematical Analysis and Applications, 389(2), 1189-1202.
• [4] Zhang, L., & Qi, L. (2016). Tensor eigenvalue complementarity problem. Journal of Mathematical Analysis and Applications, 443(2), 1245-1264.
  

Q: What is a semi-definite tensor?

A: A semi-definite tensor is a tensor that can be written as the sum of the outer products of vectors, where each vector is a non-negative multiple of a standard basis vector.

Q: What is the difference between a semi-definite tensor and a positive semi-definite tensor?

A: A positive semi-definite tensor is a semi-definite tensor that has no negative eigenvalues. In other words, a tensor $T$ is positive semi-definite if it can be written as the sum of the outer products of vectors, where each vector is a non-negative multiple of a standard basis vector, and all the eigenvalues of $T$ are non-negative.

Q: What is a convex cone?

A: A convex cone is a set of vectors that is closed under linear combinations and scaling. In other words, a set $C$ is a convex cone if it satisfies the following property: for any two vectors $x$ and $y$ in $C$, and for any non-negative real numbers $\alpha$ and $\beta$, the linear combination $\alpha x + \beta y$ is also in $C$.

Q: Is the set of all semi-definite tensors a convex cone?

A: Yes, the set of all semi-definite tensors forms a convex cone, denoted by $\mathcal{S}^n_+$. This cone is defined as the set of all tensors $T$ that can be written as the sum of the outer products of vectors, where each vector is a non-negative multiple of a standard basis vector.

Q: What are some of the properties of the convex cone of semi-definite tensors?

A: The convex cone of semi-definite tensors has several important properties, including closure under linear combinations and scaling. It is also a subset of the convex cone of positive semi-definite tensors, the convex cone of symmetric tensors, and the convex cone of non-negative tensors.

Q: What are some of the applications of semi-definite tensors?

A: Semi-definite tensors have several applications in various fields, including optimization, machine learning, and signal processing. In optimization, semi-definite tensors can be used to model the feasible region of a problem, which is the set of all possible solutions. In machine learning, semi-definite tensors can be used to model the covariance matrix of a multivariate distribution. In signal processing, semi-definite tensors can be used to model the autocorrelation matrix of a signal.

Q: How can I determine if a tensor is semi-definite?

A: To determine if a tensor is semi-definite, you can check if it can be written as the sum of the outer products of vectors, where each vector is a non-negative multiple of a standard basis vector. Alternatively, you can check if all the eigenvalues of the tensor are non-negative.

Q: What is the relationship between semi-definite tensors and positive semi-definite matrices?

A: A positive semi-definite matrix is a special case of a semi-definite tensor. In other words, a matrix $A$ is positive semi-definite if it can be written as the sum of the outer products of vectors, where each vector is a non-negative multiple of a standard basis vector, and all the eigenvalues of $A$ are-negative.

Q: Can semi-definite tensors be used in machine learning?

A: Yes, semi-definite tensors can be used in machine learning. In particular, they can be used to model the covariance matrix of a multivariate distribution, which is a fundamental concept in machine learning.

Q: What are some of the challenges associated with working with semi-definite tensors?

A: One of the challenges associated with working with semi-definite tensors is that they can be difficult to compute with, especially for large tensors. Additionally, the properties of semi-definite tensors can be complex and difficult to understand.

Q: What are some of the future directions for research on semi-definite tensors?

A: Some of the future directions for research on semi-definite tensors include developing more efficient algorithms for computing with them, and exploring their applications in machine learning and signal processing.

Q: What resources are available for learning more about semi-definite tensors?

A: There are several resources available for learning more about semi-definite tensors, including textbooks, research papers, and online courses. Some recommended resources include the book "Matrix Analysis" by Horn and Johnson, and the paper "A tutorial on tensor analysis" by Lim.

Q: What is the significance of semi-definite tensors in optimization?

A: Semi-definite tensors play a crucial role in optimization, particularly in problems involving quadratic forms. They can be used to model the feasible region of a problem, which is the set of all possible solutions.

Q: Can semi-definite tensors be used in signal processing?

A: Yes, semi-definite tensors can be used in signal processing. In particular, they can be used to model the autocorrelation matrix of a signal, which is a fundamental concept in signal processing.

Q: What are some of the applications of semi-definite tensors in finance?

A: Semi-definite tensors have several applications in finance, including modeling the covariance matrix of a portfolio of assets, and computing the value-at-risk of a portfolio.

Q: What are some of the challenges associated with working with semi-definite tensors in finance?

A: One of the challenges associated with working with semi-definite tensors in finance is that they can be difficult to compute with, especially for large portfolios. Additionally, the properties of semi-definite tensors can be complex and difficult to understand.

Q: What are some of the future directions for research on semi-definite tensors in finance?

A: Some of the future directions for research on semi-definite tensors in finance include developing more efficient algorithms for computing with them, and exploring their applications in risk management and portfolio optimization.