1. Algebraic variety
Definition
Algebraic varieties are the central objects of study in algebraic geometry.1 In the classical definition, an algebraic variety
where
Singularity (local property)
A point
A point
A singular point
Local structure: For an algebraic variety
Jacobian criterion:2 A point
Smoothness
An algebraic variety
2. Matrix variety
Definition
The set of bounded-rank matrices with
is an algebraic variety (also called matrix variety) since it can be defined by the matrices in which all
Singular points
The singular points of determinantal variety can be derived via the tools in algebraic geometry; e.g., [Theorem 10.3.3]3 or [Proposition 1.1]4. Here, we provide an elementary way to determine singularity by using the Jabobian criterion.
Given a square matrix
which leads to the full gradient
since all
which is actually a smooth manifold.
Topology
The matrix variety
3. Fixed-rank manifold
The set of all fixed-rank matrices
The matrix variety can be also interpreted by the “stratified” or “layered” space in the sense of
Tangent space
Given
where “
where a shaded square represents an arbitrary matrix and the blank represents the matrix with zero elements.
Normal space
Consider the standard Euclidean metric. The normal space at
With the help of illustrations, it is direct to see that the direct sum of tangent and normal spaces forms the Euclidean space, i.e.,
Projection
Given a matrix
4. Geometry of
Although
Tangent cone
Givenwhere
Space decomposition at singular points: Note that if
Rank increase along normal part: Another interesing fact is that given a matrix
Normal cone
GivenProjection
Given a matrix
where
Footnotes
Robin Hartshorne. “Algebraic geometry”. Vol. 52. Springer Science & Business Media, 2013.↩︎
The Jacobian criterion for testing smoothness can be found in a textbook on computational algebraic geometry, e.g., [Corollary 5.6.14] of the book: Gert-Martin Greuel, Gerhard Pfister. “A Singular introduction to commutative algebra”. Vol. 348. Berlin: Springer, 2008.↩︎
V. Lakshmibai, and Justin Brown. “The Grassmannian variety.” Developments in Mathematics. Vol. 42. Springer New York, 2015.↩︎
Winfried Bruns, and Udo Vetter. “Determinantal rings”. Vol. 1327. Springer, 2006.↩︎
John M. Lee, “Introduction to Smooth Manifolds”. Version 3.0. December 31, 2000↩︎
Nicolas Boumal. “An introduction to optimization on smooth manifolds”. Cambridge University Press, 2023.↩︎
Uri Shalit, Daphna Weinshall, and Gal Chechik. “Online learning in the manifold of low-rank matrices.” Advances in neural information processing systems 23 (2010).↩︎
Bart Vandereycken. “Low-rank matrix completion by Riemannian optimization”. In: SIAM Journal on Optimization 23.2 (2013), pp. 12141236.↩︎
Bin Gao, Renfeng Peng, Ya-xiang Yuan. “Low-rank optimization on Tucker tensor varieties.” arXiv:2311.18324 (2023).↩︎
T. P. Cason, P.-A. Absil, and P. Van Dooren, Iterative methods for low rank approximation of graph similarity matrices, Linear Algebra Appl., 438 (2013), pp. 1863–1882.↩︎
Reinhold Schneider and André Uschmajew. “Convergence results for projected line-search methods on varieties of low-rank matrices Via Lojasiewicz Inequality”. In: SIAM Journal on Optimization 25.1 (2015), pp. 622–646.↩︎
See [Section 2.1] of the work by Bin Gao, Renfeng Peng, Ya-xiang Yuan. “Low-rank optimization on Tucker tensor varieties.” arXiv:2311.18324 (2023).↩︎
Bin Gao and P.-A. Absil. “A Riemannian rank-adaptive method for low-rank matrix completion”. In: Computational Optimization and Applications 81 (2022), pp. 67–90.↩︎