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The following assumptions are made about the constant [MATH] Assumption about [MATH] (1): The set [MATH] can be covered with [MATH] |
[MATH] -balls for any [MATH] Assumption about [MATH] (2): The set [MATH] can be covered with [MATH] [MATH] -balls for any [MATH] |
All balls are spherical. A linear map from [MATH] to [MATH] can be written as [MATH] where [MATH] is the index set [MATH] [MATH] and |
[MATH] , and [MATH] is the matrix with [MATH] in the [MATH] th position if [MATH] and zero everywhere else. Here [MATH] . We use [MATH] both to refer to an entry of the vector [MATH] as in the definition of [MATH] and to the vector as a whole as in [MATH] The slight ambiguity, which is resolved from context, is highly ... |
Define [MATH] . Assume [MATH] By compactness of the ball [MATH] , we may assume the Lipshitz constant of [MATH] (with respect to |
[MATH] ) to be bounded above by [MATH] . Define [MATH] to be the set of all points [MATH] satisfying [MATH] Cover [MATH] using [MATH] balls. Suppose |
[MATH] for some [MATH] . Then by the Lipshitz bound, we must have [MATH] for [MATH] that is a center of one of the [MATH] covering [MATH] |
The rest of the argument hinges on transferring the volume [MATH] to parameter space. To do so, write [MATH] in the form [EQUATION] |
and observe that every column in the resulting matrix is in [MATH] and is all zeros except for a single entry equal to [MATH] where [MATH] denotes the projection to the [MATH] th coordinate, for some [MATH] . The first [MATH] singular values of that matrix are all equal to [MATH] . Thus, we may transfer volumes using L... |
relative to [MATH] is at most [EQUATION] By taking the limit [MATH] and because [MATH] , it follows that [MATH] for some [MATH] only for a set of [MATH] of probability zero relative to the ball [MATH] . By taking the union of the probability zero sets with [MATH] , we may conclude that [MATH] for some [MATH] |
[MATH] , only for a set of [MATH] of probability zero relative to [MATH] . Equivalently, [MATH] is injective for [MATH] with probability one relative to the ball |
[MATH] in parameter space. The argument derives its power by simply refining the cover of [MATH] by using smaller and smaller [MATH] -balls. If [MATH] |
is the tangent map at [MATH] applied to the tangent vector [MATH] , then [MATH] because of the linearity of [MATH] in [MATH] . If [MATH] is a submanifold then [MATH] is the unit tangent bundle consisting of points [MATH] with [MATH] . Injectivity may be proved by considering [MATH] instead of [MATH] with Lemma invoked ... |
Rank lemmas In proving a version of the Whitney embedding theorem, the argument of Sauer et al reviewed above writes [MATH] and relies on explicit knowledge of singular values of [MATH] . In general, singular values of [MATH] cannot be obtained so explicitly. Instead, the approach is to first argue that [MATH] has rank... |
[EQUATION] because the [MATH] th singular value [MATH] is continuous in [MATH] and [MATH] is compact. The argument may then be completed by applying Lemma |
with [MATH] To support such an argument, we give a few rank lemmas in this section. The first two lemmas are from . Rank lemmas of this type are known in multivariate approximation theory |
, although they are buried inside more sophisticated results. Suppose [MATH] . As noted already, the projection to the [MATH] th coordinate is denoted by [MATH] . If [MATH] |
[MATH] , is a multi-index, then [MATH] as usual and [MATH] . In later arguments, it is essential to take the gradient of [MATH] with respect to [MATH] . For notational convenience, we always denote [MATH] |
by [MATH] . The index set [MATH] is the set of all [MATH] such that [MATH] . By elementary combinatorics, the cardinality of [MATH] is [MATH] |
Suppose [MATH] are distinct points in [MATH] Then [EQUATION] denotes the multivariate Vandermonde matrix with the column index [MATH] for some [MATH] . The dimension of the matrix is [MATH] , where [MATH] |
is the cardinality of [MATH] Lemma 3 For [MATH] and [MATH] , the rank of the Vandermonde matrix ( 3.1 ) is equal to the number of its rows. |
Proof. Following , let [MATH] be a [MATH] orthogonal matrix drawn from the Haar measure. If [MATH] and [MATH] are distinct, then [MATH] for any [MATH] for |
[MATH] outside a set of measure [MATH] . Therefore, we can find a [MATH] such that [MATH] are distinct. We may interpolate arbitrary values at [MATH] using a univariate polynomial |
[MATH] of degree [MATH] . Because we can write [EQUATION] for a suitable choice of [MATH] , it follows that the rank of 3.1 ) is equal to the number of its rows. |
Let [EQUATION] be the multivariate incomplete Hermite matrix at [MATH] and with [MATH] Lemma 4 The rank of the incomplete Hermite matrix ( 3.2 is equal to the number of its rows if [MATH] |
Proof. Arguing as in the previous lemma, we may assume [MATH] to be distinct for [MATH] . Following and assuming [MATH] to be the identity without loss of generality, we may then find a polynomial [MATH] of degree |
[MATH] that interpolates the [MATH] th component of the prescribed gradients at [MATH] . We may then obtain the prescribed gradients from [MATH] Thus, a linear combination of the columns of ( 3.2 can produce any prescribed gradients. |
To obtain prevalence results with a fixed observation function, Lemmas and need to be combined into another lemma. Therefore, let |
[EQUATION] be the multivariate Hermite matrix at [MATH] and with [MATH] Lemma 5 The rank of the Hermite matrix ( 3.3 ) is equal to the number of its rows if [MATH] |
Proof. Suppose function values as well as gradients are prescribed at [MATH] We may obtain the prescribed gradients at [MATH] as in the previous proof in the form [MATH] To obtain suitable function values as well as the [MATH] component of the gradients, we may take the polynomial [MATH] |
with [MATH] being a suitable univariate Hermite interpolant of degree [MATH] A matrix [MATH] is said to be circulant if its subsequent rows are obtained by rotating the first row. If the number of columns is [MATH] |
and the first row is [MATH] the second row must be [MATH] The following lemma about circulant matrices will be used in the next section to refine the discussion of |
Lemma 6 Let [MATH] be a [MATH] circulant matrix whose first row is [MATH] , where [MATH] is [MATH] repeated [MATH] times. The rank of [MATH] is equal to [MATH] if [MATH] |
Proof. We must have [MATH] . Either [MATH] or [MATH] must be less than or equal to [MATH] . Because they are both integers, either [MATH] or [MATH] must be [MATH] Without loss of generality, we assume [MATH] As the rows are rotated, the [MATH] appears in column [MATH] |
for [MATH] . The columns do not wrap around because [EQUATION] All those columns are linearly independent. The final rank lemma is obvious from elementary linear algebra. We state it explicitly because it is invoked often and has a key position in the framework of |
. For the most part, the lemma is invoked silently. Lemma 7 If the rank of the matrix [MATH] is equal to the number of its rows, the rank of the product [MATH] is equal to the rank of [MATH] |
Review of Sauer et al In this section, we review the main results and proofs of Our aim is two-fold. The review helps us prepare the ground for our results about prevalence with a fixed observation map. Second, we point out and fix an error in |
, while presenting the proof with greater formal precision and completeness. The error in is a minor one relative to the depth of ideas found in that paper. We also point out errors and gaps in earlier mathematical treatments that are much more serious. |
Let [MATH] be a diffeomorphism that is at least [MATH] . We adopt the following convention: Convention about [MATH] : If [MATH] is a point in [MATH] then [MATH] [MATH] , and so on. Similarly, |
[MATH] [MATH] , and so on. It must be noted that this convention does not apply to [MATH] . For example, [MATH] are any distinct points in Lemma |
The observation function is assumed to be the (at least twice continuously differentiable) function [MATH] which maps every state vector to a real number. If the state vector is [MATH] , the corresponding delay vector is |
[EQUATION] where [MATH] will be referred to as the embedding dimension. Let [MATH] be a possibly fractal set of box counting dimension [MATH] . The set [MATH] is assumed to be compact. The delay mapping |
[MATH] restricted to [MATH] may not be injective. To examine the injectivity more generally, we perturb the observation function to |
[EQUATION] and examine injectivity in the ball [MATH] with [MATH] and fixed. The perturbed delay vector becomes [EQUATION] with [MATH] ranging over [MATH] . We use [MATH] |
instead of [MATH] to denote the delay vector for simplicity and without risk of confusion. The two assumptions about [MATH] made in the previous section are carried forward. |
Theorem 8 If [MATH] and [MATH] has finitely many periodic points [MATH] of periods less than [MATH] , the delay mapping [MATH] is injective for [MATH] for a set of [MATH] of probability |
[MATH] relative to [MATH] Theorem is less general than corresponding statements in . Our aim is to exhibit techniques while forsaking generality. The manner in which more general statements can be obtained is discussed later. |
Proof. Define [MATH] We then have [MATH] where [EQUATION] Here [MATH] is [MATH] and [MATH] is [MATH] where [MATH] is the cardinality of [MATH] . The proof turns on the determination of the rank of [MATH] . If |
[MATH] and [MATH] [MATH] , are [MATH] distinct points, we may apply Lemmas and and immediately conclude that the rank of [MATH] is [MATH] However, if not all points are distinct, the rank of [MATH] |
is obviously not equal to the number of rows. Several cases need to be considered to determine the rank of [MATH] Case 1: both [MATH] and [MATH] are periodic of period less than [MATH] with [MATH] . The set of such pairs [MATH] |
is finite (by assumption) and will be denoted by [MATH] There are two subcases. Case 1.1: [MATH] and [MATH] lie on distinct orbits. If so [MATH] can be written in a compressed form as [MATH] |
with [EQUATION] where [MATH] are the periods of [MATH] (or [MATH] if the periods are greater than [MATH] ), respectively. Further, [MATH] |
is a [MATH] circulant matrix with first row [MATH] and [MATH] is a [MATH] circulant matrix with first row [MATH] . The rank of [MATH] is equal to the number of its rows by Lemma and [MATH] |
is nonzero. Therefore, we may assert that the rank of [MATH] is [MATH] or greater. Case 1.2: [MATH] and [MATH] lie on the same periodic orbit. In this case, we may write |
[EQUATION] where [MATH] is the period of [MATH] [MATH] is a [MATH] circulant matrix whose first row is of the form [MATH] Again, we conclude that the rank of [MATH] is greater than |
[MATH] Suppose [MATH] for some [MATH] Then [MATH] and [MATH] must lie on a hyperplane of co-dimension [MATH] or greater. Because [MATH] is finite, we may assert [MATH] for all [MATH] |
with probability [MATH] relative to the ball [MATH] Case 1 is now complete. Case 2: Define [MATH] to be the set of all [MATH] satisfying |
1. [MATH] 2. [MATH] All distances in this paper use the [MATH] or spectral norm. The matrix [MATH] has a rank equal to [MATH] for each point in [MATH] , as we will prove by breaking up case 2 into subcases. |
Case 2.1: [MATH] are [MATH] distinct points. In this case, [MATH] has rank equal to [MATH] as noted at the beginning of the proof. |
Case 2.2: [MATH] are distinct, [MATH] are distinct, and neither [MATH] nor [MATH] is a periodic point of period less than [MATH] , but [MATH] or [MATH] for |
[MATH] . Without loss of generality, we assume [MATH] In this case, the compressed form is [MATH] with [EQUATION] where [MATH] is [MATH] circulant matrix with first row equal to [MATH] . The [MATH] does not wrap around and the rank of [MATH] and therefore of [MATH] is [MATH] |
Case 2.3: [MATH] periodic of period less than [MATH] and [MATH] not so (or vice versa, which may be ignored without loss of generality). In this case, the compressed form is [MATH] |
with [EQUATION] where [MATH] is the period of [MATH] [MATH] is a [MATH] circulant matrix with first row [MATH] , and [MATH] is a [MATH] |
circulant matrix with first row [MATH] The column rank of [MATH] is equal to [MATH] and therefore the rank of [MATH] is also [MATH] |
We can now complete case 2 as follows. Suppose [MATH] for some [MATH] . By assumption (2) about [MATH] , cover [MATH] with [MATH] |
or fewer [MATH] -balls for [MATH] . At this point, we introduce an assumption about [MATH] Assumption about [MATH] (1): The Lipshitz constant of [MATH] |
with respect to [MATH] and with [MATH] is bounded by [MATH] . The existence of [MATH] is a consequence of the compactness of [MATH] , the compactness of [MATH] and the differentiability assumption about the observation function |
[MATH] and the diffeomorphism [MATH] It then follows that if [MATH] at some point [MATH] , then [MATH] at the center of one of the [MATH] -balls covering [MATH] Define |
[EQUATION] By compactness of [MATH] [MATH] exists and is positive. By the transfer of volume Lemma which is applied with [MATH] , the probability of [MATH] relative to the ball [MATH] at a point [MATH] |
is upper bounded by [EQUATION] Because [MATH] can be covered with [MATH] or fewer [MATH] -balls, the probability that [MATH] for some [MATH] is upper bounded by |
[EQUATION] Because [MATH] and by taking [MATH] , we conclude that the probability of [MATH] for some [MATH] relative to [MATH] is one. Case 2 is now complete. |
To complete the proof of injectivity, take the union of the measure zero sets in case 2 with [MATH] and the measure zero set in case 1. Outside of that measure [MATH] subset of the ball [MATH] , we have [MATH] |
for [MATH] and [MATH] The ideas in the proof presented above are from although our presentation is more precise and formally complete. Theorem |
makes an assumption on periodic points of period [MATH] and not [MATH] as in To see why the more stringent assumption is needed, we turn to , p. 611, case 3] The case “ [MATH] and [MATH] are not both periodic with period [MATH] is considered ( [MATH] is [MATH] in our notation) and it is stated that |
[MATH] (which is [MATH] in our notation) is triangular of rank [MATH] . Unfortunately, that statement is not correct. To understand why that statement is not true, assume [MATH] . Suppose |
[MATH] is a periodic point of period [MATH] and that [MATH] Then [MATH] will be a [MATH] circulant matrix which looks as follows: |
[EQUATION] Evidently, the rank of this matrix is [MATH] The easiest way to fix the minor error is to assume the number of periodic points of period [MATH] to be finite as we have done. However, Sauer et al |
place conditions on the box counting dimension of the set of periodic points of period [MATH] The conditions involving quantities such as [MATH] |
are not easy to interpret and it is unclear what they mean. The basic idea of assuming a bound on the box counting dimension of periodic points of a certain period is a sound one. It can be developed fully using Lemma about the rank of circulant matrices and variations of that lemma. We have not done so for two reasons... |
[MATH] , then [MATH] will be a characteristic multiplier that is repeated more than once, which is excluded in the immersivity theorem. |
The gaps in and are much less minor. In , it is assumed that the delay map is an embedding in some neighborhood of the periodic points. The proof of that assumption is unlikely to be as straightforward as assumed. Even granting that assumption, the argument for transversality 10 , p. 371] |
appears incomplete. In particular, it does not consider the possibility that perturbing the delay map of [MATH] may also perturb the delay map of [MATH] , for example, when [MATH] and the orbits of [MATH] |
overlap. There are yet other aspects of the proof we were not able to verify. For example, 10 , p. 370, case iii] seems to require [MATH] to be close to a periodic point and [MATH] to be away from a periodic point. It is then asserted that [MATH] |
are distinct. How could that be true if [MATH] is a fixed point? How is the possibility [MATH] handled? The gaps in also occur in handling overlaps of orbits and periodic points. The main argument , p. 598] |
entirely ignores the possibility that orbits of [MATH] and [MATH] may overlap. Further, it is suggested that difficulties associated with fixed points can be handled by adjusting the delays but no details are provided about carrying out that suggestion. |
Going back to the work of Sauer et al a point in our proof of Theorem is worth calling to attention. In the proof, [MATH] is covered with [MATH] -balls and it is assumed that every ball center is in [MATH] . It is not sufficient to start with any cover of [MATH] because a ball center can be arbitrarily close to the dia... |
may become arbitrarily small. If we say that a certain compact set [MATH] is covered by a certain number of [MATH] -balls, it is assumed that each ball has a center that lies in [MATH] . That assumption comes up repeatedly in the proof of immersivity, which we now turn to. Once again all the ideas are from |
. Here [MATH] is assumed to be a smooth, closed, and compact submanifold of dimension [MATH] and [MATH] denotes its unit tangent bundle. If [MATH] |
and [MATH] is tangent to [MATH] at [MATH] , then [MATH] if and only if [MATH] Theorem 9 If [MATH] [MATH] is invariant under [MATH] [MATH] has finitely many points [MATH] of period less than [MATH] , and all characteristic multipliers of each of those points are distinct, then [MATH] is immersive over |
[MATH] with probability [MATH] relative to the ball [MATH] Proof. If [MATH] and [MATH] is a tangent vector to [MATH] at [MATH] , then we denote the vector that [MATH] is mapped to by [MATH] The following convention about [MATH] is an extension of the convention about [MATH] explained earlier. |
Convention about [MATH] : If [MATH] is tangent to [MATH] at [MATH] then [MATH] [MATH] , and so on. Because [MATH] is a diffeomorphism, [MATH] are all nonzero like [MATH] |
We write [MATH] where [EQUATION] The proof will turn on the rank of [MATH] . If [MATH] are distinct, the rank of [MATH] is [MATH] because the rank of [MATH] is equal to the number of its rows by Lemma and the rank of |
[MATH] is obviously [MATH] To study the rank of [MATH] , it is useful to define the following disjoint sets of [MATH] [MATH] is the set of all [MATH] such that [MATH] |
is a periodic point of period less than [MATH] and [MATH] is an eigenvector of the periodic point [MATH] . By eigenvector of a periodic point, we mean an eigenvector of the corresponding monodromy matrix. |
[MATH] is the set of all [MATH] such that [MATH] is a periodic point of period less than [MATH] and [MATH] is a linear combination of two eigenvectors of [MATH] . It is also required that |
[EQUATION] We will denote [MATH] , where this last condition is not operative, by [MATH] . Evidently, [MATH] is a subset of [MATH] |
In general, [MATH] , where [MATH] , is defined as the set of [MATH] such that [MATH] is a periodic point of period [MATH] or less and [MATH] is a linear combination of [MATH] eigenvectors of the periodic point [MATH] .It is also required that |
This sequence of cases stops at [MATH] and does not go up to [MATH] because we are only interested in those eigenvectors of the periodic point [MATH] that are also tangent to [MATH] . The assumption about the invariance of [MATH] is used here. |
The final case is [MATH] which consists of all points [MATH] such that [MATH] is not periodic of period less than [MATH] and the distance to [MATH] is |
[MATH] The final case [MATH] is the easiest to handle. In this case, [MATH] are distinct and the rank of [MATH] is [MATH] as already mentioned. |
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