Abstract
The sciences do not try to explain, they hardly even try to interpret, they mainly make models. By a model is meant a mathematical construct which, with the addition of certain verbal interpretations, describes observed phenomena. The justification of such a mathematical construct is solely and precisely that it is expected to work.
The sciences do not try to explain, they hardly even try to interpret, they mainly make models. By a model is meant a mathematical construct which, with the addition of certain verbal interpretations, describes observed phenomena. The justification of such a mathematical construct is solely and precisely that it is expected to work.
—John von Neumann
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Notes
- 1.
Under fairly broad conditions that we will elaborate in this book.
- 2.
In the statistical learning literature, the most commonly used term is “mixture model.” In systems theory, the typical term is “hybrid model.” In algebraic geometry, for the case of subspaces, the typical term is a “subspace arrangement.”
- 3.
In this book, we will use interchangeably “mixture,” “collection,” “union,” and “arrangement” of subspaces or models. But be aware that in the case of subspaces, the formal terminology in algebraic geometry is a “subspace arrangement.”
- 4.
Or equivalently, we may impose a nonuniform prior distribution over all models.
- 5.
Occam’s (or Ockham’s) razor is a principle attributed to the fourteenth-century logician and Franciscan friar William of Occam: “Pluralitas non est ponenda sine neccesitate,” which translates literally as “entities should not be multiplied unnecessarily.” In science, this principle is often interpreted thus: “when you have two competing theories that make exactly the same predictions, the simpler one is better.”
- 6.
Roughly speaking, a smooth manifold is a special topological space that is locally homeomorphic to a Euclidean space and has the same dimension everywhere. A general topological space may have singularities and consist of components of different dimensions.
- 7.
If the true distribution from which the data are drawn is \(q(\boldsymbol{x})\), then the maximum likelihood estimate \(p(\boldsymbol{x}\mid \theta ^{{\ast}})\) minimizes the Kullback–Leibler (KL) divergence \(KL(q\|p) =\int q(\boldsymbol{x})\log \frac{q(\boldsymbol{x})} {p(\boldsymbol{x})}\ d\boldsymbol{x}\) among the given class of distributions (see Appendix B.)
- 8.
Singular distributions are probability distributions concentrated on a set of Lebesgue measure zero. Such distributions are not absolutely continuous with respect to the Lebesgue measure. The Cantor distribution is one example of a singular distribution.
- 9.
As the cluster centers.
- 10.
An object is called Lambertian if its apparent brightness is the same from any viewpoint.
- 11.
Depending on the illumination model, the illumination space can be approximately three- or nine-dimensional.
- 12.
The legitimacy of the projection process will be addressed in Chapter 5
- 13.
In contrast to the previous face example, there is no rigorous mathematical justification for why local profiles from a region of similar texture must span a low-dimensional linear subspace. However, there is strong empirical evidence that a linear subspace normally gives a very good approximation.
- 14.
That is, the number d i of basis elements needed to represent the ith region is typically much smaller than the number d of basis elements needed to represent the whole image.
- 15.
An algebraic variety is an irreducible algebraic set. An algebraic set is called irreducible if it cannot be written as the union of two proper algebraic subsets. A subspace is one such example.
- 16.
A prime ideal is an ideal that cannot be decomposed further as the intersection of two other ideals (see Appendix C). The zero-level set of a prime ideal is an irreducible algebraic set, i.e., an algebraic variety.
- 17.
Notice the correspondence between a “union” of algebraic varieties and the “multiplication” of the polynomials associated with the varieties.
- 18.
According to Hilbert’s Nullstellensatz (see Appendix C), there is a one-to-one correspondence between algebraic sets and radical ideals (Eisenbud 1996).
- 19.
For the special case in which the ideal is generated by a single polynomial, the decomposition is equivalent to factoring the polynomial into factors.
- 20.
For example, every function can be approximated arbitrarily well by a piecewise linear function with a sufficient number of pieces.
- 21.
For instance, the complexity of a model can be measured as the minimum number of bits needed to fully describe the model, and the data fidelity can be measured by the distance from the sample points to the model.
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Vidal, R., Ma, Y., Sastry, S.S. (2016). Introduction. In: Generalized Principal Component Analysis. Interdisciplinary Applied Mathematics, vol 40. Springer, New York, NY. https://doi.org/10.1007/978-0-387-87811-9_1
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