Tight frames

Abstract

Decompositions like those in our two prototypical examples, i.e., $f = \sum _{j\in J} \langle f, f_j\rangle f_j, \qquad \forall f\in {\mathscr {H}},$ will come from what is called a tight frame $(f_j)_{j\in J}$.

Appendices

Notes

The key idea (not to be underestimated) of this section is:

Tight frames are best understood via their Gramian.

Indeed, a tight frame $\varPhi =(f_j)_{j=1}^n$ is determined up to unitary equivalence (and normalisation) by its Gramian $P=P_\varPhi$, which is an orthogonal projection matrix. The columns of $P_\varPhi$ give a (canonical) copy of $\varPhi$, and so the kernel of $P_\varPhi$ is the space of linear dependencies between the vectors in $\varPhi$, i.e.,

$$ \ker (P_\varPhi ) =\{a\in \mathbb {F}^n:Pa=\sum _j a_j Pe_j=0\} =\{a\in \mathbb {F}^n:\sum _j a_j f_j=0\} =:\mathop {\mathrm{dep}}\nolimits (\varPhi ) . $$

Since $P_\varPhi$ is determined by $\ker (P_\varPhi )$, this observation allows the theory of tight frames to be extended to any finite dimensional vector space over a subfield of $\mathbb {C}$ which is closed under conjugation (see Chapter 4).

Many notions of equivalence of tight frames appear in the literature (see [Bal99], [HL00], [GKK01], [Fic01], [HP04]). Here we use a descriptive terminology (from which all of these can be described). For finite tight frames viewed as sequences of vectors, unitary equivalence is the natural equivalence, and when viewed as (weighted) projections (fusion frames), projective unitary equivalence is natural. Unitary equivalence is determined by the Gramian (Corollary 2.1), and projective unitary unitary equivalence is determined by certain m-products (see Chapter 8).

It is implicit in the Definition 2.1 of a tight frame that ${\mathscr {H}}$ be separable, i.e., have a countable orthonormal basis. The theory extends, in the obvious way, to nonseparable spaces, with J now an uncountable index set. In these cases, it turns out that all tight frames for ${\mathscr {H}}$ (with nonzero vectors) have the same infinite cardinality, i.e., the Hilbert dimension of ${\mathscr {H}}$. By way of contrast, if ${\mathscr {H}}$ has finite dimension d, then there exist tight frames for ${\mathscr {H}}$ with any countable cardinality greater than or equal to d.

We will have good reason to consider representations such as (1.3), where the sum $\sum _{j\in J}$ is replaced by a continuous sum (with respect to some measure). This generalisation (see Chapter 16) will be called a continuous tight frame, with the special case of Definition 2.1 referred to as a (discrete) tight frame.

The book [HKLW07] covers the material of this section. It has a section on frames in $\mathbb {R}^2$ (for tight frames in $\mathbb {R}^3$ see [Fic01]). The popular article [KC07a, KC07b] advocates the use of tight frames in a number of engineering applications. It outlines standard terminology for frames (resulting from an e-mail discussion within the frame community), which we adopt, except for our preference of normalised tight frame over Parseval frame. In this parlance a ENPTF is a equal-norm Parseval tight frame, and similarly.

Exercises

2.1.

By expanding, or otherwise, verify the polarisation identity for an inner product space ${\mathscr {H}}$, i.e., $\forall f, g\in {\mathscr {H}}$ that

2.2.

Use the polarisation identity to show that the following conditions are equivalent to being a finite tight frame

2.3.

Orthogonal projection formula.

Let $(f_j)_{j\in J}$ be a finite tight frame (with frame bound A) for a subspace ${\mathscr {K}}\subset {\mathscr {H}}$. Show that P the orthogonal projection onto this subspace is given by

$$ P={1\over A}VV^*: f\mapsto {1\over A}\sum _{j\in J}\langle f, f_j\rangle f_j, \qquad V:=[f_j]_{j\in J}. $$

2.4.

Orthogonal bases and tight frames.

(a) Show that an orthogonal basis $(f_j)_{j\in J}$ for ${\mathscr {H}}$ is a tight frame if and only if all its vectors have the same norm and that it is a normalised tight frame if and only if it is an orthonormal basis.

(b) Show that if $(f_j)$ is a normalised tight frame, then $\Vert f_j\Vert \le 1$, $\forall j\in J$, and

$$ \Vert f_j\Vert =1 \qquad \iff \qquad f_j\perp f_k,\quad \forall k\ne j. $$

In particular, the only normalised tight frames whose vectors all have unit length are the orthonormal bases.

2.5.

Unitary images of tight frames.

(a) Show that the image of a tight frame $(f_j)_{j\in J}$ under a unitary map U is a tight frame with the same frame bound.

(b) Show that if $(f_j)$ is a finite normalised tight frame for ${\mathscr {H}}$, and T is a linear map for which $(Tf_j)$ is also, then T is a unitary map.

2.6.

Projections of normalised tight frames are normalised tight frames.

A linear map $P:{\mathscr {H}}\rightarrow {\mathscr {H}}$ on a Hilbert space is an orthogonal projection if $P^2=P$ and $P^*=P$. Show that if $(f_j)_{j\in J}$ is a normalised tight frame for a Hilbert space ${\mathscr {K}}$ and P is an orthogonal projection onto a subspace ${\mathscr {H}}\subset {\mathscr {K}}$, then $(Pf_j)$ is a normalised tight frame for ${\mathscr {H}}$. (This is obvious in the context of Theorem 2.2.)

2.7.

Partial isometries map normalised tight frames to normalised tight frames.

A linear map $L:{\mathscr {H}}\rightarrow {\mathscr {K}}$ between Hilbert spaces is an isometry if $L^*L=I_{\mathscr {H}}$, i.e., it is norm preserving:

$$ \Vert Lx\Vert =\Vert x\Vert , \qquad \forall x\in {\mathscr {H}}. $$

It is a coisometry if $L^*:{\mathscr {K}}\rightarrow {\mathscr {H}}$ is an isometry, i.e., $L^*$ is norm preserving.

Let $\varPhi$ be a finite normalised tight frame for ${\mathscr {H}}$, and $Q:{\mathscr {H}}\rightarrow {\mathscr {K}}$ be a linear map. Show that the following are equivalent

(a) Q is a partial isometry, i.e., its restriction to $(\ker Q)^\perp =\mathop {\mathrm{ran}}\nolimits (Q^*)$ is an isometry.

(b) $QQ^*$ is an orthogonal projection.

(c) $Q^*Q$ is an orthogonal projection.

(d) $Q\varPhi$ is a normalised tight frame (for its span).

Remark. Since unitary maps and orthogonal projections are partial isometries, this generalises Exercises 2.5 and 2.6. It appears as a special case in Exer. 3.5.

2.8.

$^\mathtt{m}$If U is an $n\times n$ unitary matrix (or a scalar multiple of one) with entries of constant modulus, then an equal-norm tight frame for $\mathbb {F}^d$ is given by the columns of the $d\times n$ submatrix obtained from it by selecting any d of its rows.

(a) When $\mathbb {F}=\mathbb {R}$, such U, with entries $\pm 1$, are called Hadamard matrices. Use the matlab function hadamard(n) (defined for n,${n\over 12}$ or ${n\over 20}$ a power of 2) to construct equal-norm tight frames of n vectors in $\mathbb {R}^d$.

Remark: It can be shown that if a Hadamard matrix exists, then $n=1,2$ or n is divisible by 4. The Hadamard conjecture is that there exists a Hadamard matrix of size $n=4k$, for every k. The smallest open case (in 2010) is $n=668$.

(b) Show that the Fourier matrix $F= {1\over \sqrt{n}} [\omega ^{jk}]_{0\le j, k<n}$, $\omega =e^{2\pi i\over n}$ of (2.5) is unitary and has order 4. Use the matlab function fft(X) (Discrete Fourier transform) to construct F, and hence equal-norm tight frames of n vectors in $\mathbb {C}^d$.

Remark: It is always possible to obtain a real frame in this way.

2.9.

Tight frames for $\mathbb {R}^2$.

(a) Show the vectors $(v_j)_{j=1}^n$ , $v_j=(x_j, y_j)\in \mathbb {R}^2$ are a tight frame for $\mathbb {R}^2$ if and only if the diagram vectors $w_j := (x_j+i y_j)^2\in \mathbb {C}$ sum to zero (in $\mathbb {C}$).

(b) Show that two tight frames for $\mathbb {R}^2$ are projectively unitarily equivalent if and only if their diagram vectors are scalar multiples of each other.

(c) Show that up to projective unitary equivalence the only equal-norm tight frame of three vectors for $\mathbb {R}^2$ is three equally spaced unit vectors.

(d) Show that all unit-norm tight frames of four vectors for $\mathbb {R}^2$ are the union of two orthonormal bases. This gives a one-parameter family of projectively unitarily inequivalent unit-norm tight frames of four vectors for $\mathbb {R}^2$.

(e) Show the tight frames of five unit vectors for $\mathbb {R}^2$ with diagram vectors

$$ \{e^{i\theta },e^{-i\theta },e^{i(\theta +\psi )}, e^{-i(\theta +\psi )},-1\}, \qquad 0<\theta <{\pi \over 2}, \quad \cos 2\theta +\cos (2\theta +2\psi )={1\over 2}$$

are projectively unitarily inequivalent, and that none is the union of an orthonormal basis and three equally spaced vectors.

2.10.

Projective unitary equivalence in $\mathbb {R}^2$.

(a) For unit-norm tight frames of n vectors for $\mathbb {R}^2$ show that the equivalence classes for projective unitary equivalence up to reordering are in 1–1 correspondence with convex n-gons with sides of unit length (given by a sum of diagram vectors).

(b) What do subsets of orthonormal vectors correspond to on the polygon?

(c) What is the n-gon corresponding to the tight frame for $\mathbb {R}^2$ given by n equally spaced unit vectors?

(d) Does every finite tight frames for $\mathbb {R}^2$ correspond to some convex polygon?

2.11.

The complex conjugate of ${\mathscr {H}}$ is the Hilbert space $\overline{{\mathscr {H}}}$ of all formal complex conjugates with addition, scalar multiplication and inner product given by

$$\begin{aligned} \overline{v}+\overline{w} = \overline{v+w},\qquad \alpha \overline{v}=\overline{\overline{\alpha }v}, \qquad \langle \overline{v},\overline{w}\rangle = \overline{\langle v, w\rangle }. \end{aligned}$$

(2.18)

(a) Show that the conjugation map $C:{\mathscr {H}}\rightarrow \overline{{\mathscr {H}}}:v\mapsto \overline{v}$ is antilinear.

(b) Suppose that $\varPhi =(f_j)$ is a sequence of vectors in ${\mathscr {H}}$ and $\overline{\varPhi }:=(\overline{f_j})\subset \overline{{\mathscr {H}}}$. Show that the frame operator and Gramian satisfy

$$ S_{\overline{\varPhi }}=C S_\varPhi C^{-1}, \qquad \mathop {\mathrm{Gram}}\nolimits (\overline{\varPhi })=\overline{\mathop {\mathrm{Gram}}\nolimits (\varPhi )}. $$

Hence C maps tight frames to a tight frames (with the same bound A).

(c) Suppose that ${\mathscr {H}}=V$, with V a subspace of $\mathbb {C}^d$. Show that $\overline{{\mathscr {H}}}$ is isomorphic to the subspace $\overline{V}:=\{\overline{v}:v\in V\}$ of $\mathbb {C}^d$, where $\overline{v}=\overline{(v_j)}:=(\overline{v_j})$.

2.12.

Show the frame operator S for a sequence of vectors $f_1,\ldots , f_n$ satisfies:

(a) $\mathop {\mathrm{trace}}\nolimits (S)=\sum _j \Vert f_j\Vert ^2$.

(b) $\mathop {\mathrm{trace}}\nolimits (S^2)=(\Vert S\Vert _F)^2 = \sum _j \sum _k |\langle f_j, f_k\rangle |^2$.

Hint: The trace operator satisfies $\mathop {\mathrm{trace}}\nolimits (AB)=\mathop {\mathrm{trace}}\nolimits (BA)$.

2.13.

Trace formula. Show that if $(f_j)_{j\in J}$ is a finite normalised tight frame for ${\mathscr {H}}$ and $L:{\mathscr {H}}\rightarrow {\mathscr {H}}$ is a linear transformation, then its trace is given by

$$ \mathop {\mathrm{trace}}\nolimits (L)=\sum _{j\in J}\langle Lf_j, f_j\rangle . $$

In particular, when L is the identity map, we obtain the trace formula (2.9).

2.14.

Let ${\mathscr {H}}$ be have finite dimension $d\ge 1$. Show that

(a) There exists a tight frame $(f_j)_{j=1}^\infty$ for ${\mathscr {H}}$, with infinitely many nonzero vectors.

(b) For any such tight frame, $\Vert f_j\Vert \rightarrow 0$ as $j\rightarrow \infty$.

(c) There are no equal-norm tight frames for ${\mathscr {H}}$ with infinitely many vectors.

Remark: In contrast, the continuous tight frame $(u_\theta )$ for $\mathbb {R}^2$ of (1.3) has uncountably many equal-norm vectors.

2.15.

Equal-norms. Show that if $(f_j)$ is an equal-norm tight frame of n vectors (with frame bound A) for a space ${\mathscr {H}}$ of dimension d, then

$$ \Vert f_j\Vert =\sqrt{dA\over n}, \qquad \forall j. $$

In particular, if $(f_j)$ is unit-norm, i.e., $\Vert f_j\Vert =1$, $\forall j$, then ${1\over A}={d\over n}$,

2.16.

Equiangularity. Show that if $(f_j)$ is an equiangular tight frame of $n>1$ vectors (with frame bound A) for a space ${\mathscr {H}}$ of dimension d, then its Gramian satisfies

$$ \langle f_j,f_j\rangle ={dA\over n}, \quad \forall j, \qquad |\langle f_j, f_k\rangle | = {A\over n}\sqrt{d(n-d)\over n-1}, \quad j\ne k. $$

2.17.

Let $\varPhi =(f_j)_{j\in J}$ be a finite sequence of vectors in ${\mathscr {H}}$, where $d=\dim ({\mathscr {H}})$, and $V:=[f_j]_{j\in J}$. Show $\varPhi$ is a normalised tight frame for ${\mathscr {H}}$ if and only if

(a) $\mathop {\mathrm{Gram}}\nolimits (\varPhi )=V^*V$ has exactly d nonzero eigenvalues all equal to 1.

(b) The frame operator $S_\varPhi =VV^*$ has all its eigenvalues equal to 1.

(c) The synthesis operator V has d singular values equal to 1.

(d) The analysis operator $V^*$ has d singular values equal to 1.

2.18.

Isometries. Let $\varPhi =(f_j)_{j\in J}\subset {\mathscr {H}}$, and $V^*:{\mathscr {H}}\rightarrow \ell _2(J) :f\mapsto (\langle f, f_j\rangle )_{j\in J}$ be the analysis operator. Show that the following are equivalent

(a) $\varPhi$ is a normalised tight frame for ${\mathscr {H}}$.

(b) $V^*$ is inner product preserving, i.e., $\langle V^*f,V^*g\rangle =\langle f, g\rangle$, $\forall f, g\in {\mathscr {H}}$.

(c) $V^*$ is an isometry, i.e., $\Vert V^*f\Vert =\Vert f\Vert$, $\forall f\in {\mathscr {H}}$.

2.19.

Suppose that $\varPhi =(f_j)_{j=1}^n$ and $\varPsi =(g_j)_{j=1}^n$ are sequences of vectors, with ${\mathscr {H}}=\mathop {\mathrm{span}}\nolimits (\varPhi )$ and ${\mathscr {K}}=\mathop {\mathrm{span}}\nolimits (\varPsi )$. Show there is a unitary map $U:{\mathscr {H}}\rightarrow {\mathscr {K}}$ with $g_j=Uf_j$, $\forall j$ if and only if $\langle f_j,f_k\rangle =\langle g_j, g_k\rangle$, $\forall j, k$, i.e., $\mathop {\mathrm{Gram}}\nolimits (\varPhi )=\mathop {\mathrm{Gram}}\nolimits (\varPsi )$.

2.20.

(a) Express unitary equivalence up to reordering in terms of the Gramian.

(b) Express projective unitary equivalence up to reordering in terms of the Gramian.

(c) Show that a necessary, but not sufficient, condition for normalised tight frames $(f_j)_{j\in J}$ and $(g_j)_{j\in K}$ to be projectively equivalent up to reordering is that there is a permutation $\sigma :J\rightarrow K$ with $|\langle g_{\sigma j},g_{\sigma k}\rangle |=|\langle f_j, f_k\rangle |$, $\forall j, k\in J$. In particular, the multisets $\{ |\langle f_j,f_k\rangle |\}_{j, k\in J}$ and $\{ |\langle g_j,f_k\rangle |\}_{j, k\in J}$ must be equal.

2.21.

(a) Show that normalised tight frames $\varPhi$ and $\varPsi$ are projectively unitarily equivalent up to reordering if and only if their complements are.

(b) Show that all equal-norm tight frames of $n=d+1$ vectors in $\mathbb {F}^d$ are projectively unitarily equivalent, and hence are equiangular.

(c) For the unitarily inequivalent equal-norm frames of three vectors for $\mathbb {C}^2$ given in Example 2.8, find a unitary matrix U (and scalars $\alpha _j$) which gives the projective unitary equivalence $g_j=\alpha _j U f_j$, where $\varPhi =(f_j)$ and $\varPsi =(g_j)$.

2.22.

Find all possible normalised tight frames of three vectors for $\mathbb {C}^2$ up to unitary equivalence.

2.23.

Show that no tight frame can be unitarily equivalent to its complement. Can a tight frame be projectively unitarily equivalent to its complement?

2.24.

$^\mathtt{m}$ Write a matlab function for the complementary tight frame using null.

2.25.

$^\mathtt{m}$ (a) By using an inductive argument based on complements, prove that an equal-norm tight frame of n vectors for $\mathbb {F}^d$ can be constructed, for all $n>d$.

(b) Write a function ENTF(n, d) to construct such equal-norm tight frames.

(c) Construct an equal-norm tight frame of 8 vectors for $\mathbb {R}^3$.

2.26.

M. A. Naĭmark’s theorem.

An orthogonal resolution of the identity for a Hilbert space ${\mathscr {H}}$ is a one-parameter family $(E_t)_{t\in \mathbb {R}}$ of orthogonal projections on ${\mathscr {H}}$, for which $t\mapsto E_t$ is left continuous, and

$$ \lim _{t\rightarrow -\infty } E_t=0,\quad \lim _{t\rightarrow \infty } E_t = I_{\mathscr {H}}, \qquad E_sE_t = E_{\min \{s, t\}}. $$

A generalised resolution of the identity is a family $(F_t)_{t\in \mathbb {R}}$, for which the differences $F_t-F_s$, $s<t$ are bounded positive operators, $t\mapsto F_t$ is left continuous, and

$$ \lim _{t\rightarrow -\infty } F_t=0,\quad \lim _{t\rightarrow \infty } F_t = I_{\mathscr {H}}. $$

Naĭmark’s theorem (see, e.g., [AG63]) says that every generalised resolution of the identity for ${\mathscr {H}}$ is the orthogonal projection onto ${\mathscr {H}}$ of an orthogonal resolution of the identity for some larger Hilbert space ${\mathscr {K}}\supset {\mathscr {H}}$.

(a) Let $(f_j)_{j=1}^n$ be a finite normalised tight frame for which none of the vectors are zero. Show that a generalised resolution of the identity is given by

$$ F_t f := \sum _{j\le t}\langle f, f_j\rangle f_j, \qquad \forall f\in {\mathscr {H}}.$$

(b) By Naĭmark’s theorem, there is a Hilbert space ${\mathscr {K}}\supset {\mathscr {H}}$, and an orthogonal resolution of the identity $(E_t)$ for ${\mathscr {K}}$, such that $F_t = P E_t$, where P is the orthogonal projection of ${\mathscr {K}}$ onto ${\mathscr {H}}$. Conclude that

$$ I_{\mathscr {H}}= \sum _{j=1}^n (F_j-F_{j-1}) = \sum _{j=1}^n P Q_j, \qquad Q_j:=E_j-E_{j-1} $$

where $Q_j$ is an orthogonal projection, and $Q_j\perp Q_k$, $k\ne j$.

(c) Show that ${\mathscr {K}}$ can be taken to be n dimensional.

(d) Prove Naĭmark’s theorem for ${\mathscr {H}}=\mathbb {F}^d$ by taking $V=[f_1,\ldots , f_n]$ which has orthonormal rows, and extend it to obtain a unitary matrix.

2.27.

Suppose that $(u_j+iv_j)_{j=1}^n$ is a normalised tight frame of n vectors for $\mathbb {C}^d$, where $u_j, v_j\in \mathbb {R}^d$. Prove that $(u_1,\ldots ,u_n, v_1,\ldots , v_n)$ is a normalised tight frame of 2n vectors for $\mathbb {R}^d$.

2.28.

Show that a tight frame for $\mathbb {R}^d$ is tight frame for $\mathbb {C}^d$.

2.29.

Normalised tight frames and linear mappings.

Let $(f_j)_{j\in J}$ and $(g_k)_{k\in K}$ be finite normalised tight frames for ${\mathscr {H}}$ and ${\mathscr {K}}$. Denote the vector space of all linear maps ${\mathscr {H}}\rightarrow {\mathscr {K}}$ by ${\mathscr {L}}({\mathscr {H}},{\mathscr {K}})$.

(a) Show that the Hilbert–Schmidt inner product on ${\mathscr {L}}({\mathscr {H}},{\mathscr {K}})$ satisfies

$$ \langle L,M\rangle _{HS} := \mathop {\mathrm{trace}}\nolimits (M^*L) = \sum _{j\in J} \langle Lf_j,Mf_j\rangle = \sum _{k\in K} \langle M^*g_k, L^*g_k\rangle . $$

Remark: Taking $M=I_{\mathscr {H}}$ gives the trace formula of Exer. 2.13.

(b) Let $f_j^*$ be ${\mathscr {H}}\rightarrow \mathbb {F}:f\mapsto \langle f, f_j\rangle$. Show that $(g_k f_j^*)_{j\in J, k\in K}$ is a normalised tight frame (of rank one maps) for ${\mathscr {L}}({\mathscr {H}},{\mathscr {K}})$ with the Hilbert–Schmidt inner product.

2.30.

Matrices with respect to a normalised tight frame.

Normalised tight frames can be used to represent vectors and linear maps in much the same way as orthonormal bases. Suppose that $(f_j)_{j\in J}$ and $(g_k)_{k\in K}$ are finite normalised tight frames for ${\mathscr {H}}$ and ${\mathscr {K}}$, and let $V=[f_j]_{j\in J}$, $W=[g_k]_{k\in K}$. Then the coordinates x of $f\in {\mathscr {H}}$ with respect to $(f_j)$, and the matrix A representing a linear map $L:{\mathscr {H}}\rightarrow {\mathscr {K}}$ with respect to $(f_j)$ and $(g_k)$ are

$$ x=[f]:=V^*f\in \mathbb {F}^J, \qquad A=[L]:=W^*LV\in \mathbb {F}^{J\times J}. $$

(a) Show that $[Lf]=Ax$, and f, L can be recovered via $f=Vx$, $L=WAV^*$.

(b) Show that $[\alpha L+\beta M]=\alpha [L]+\beta [M]$, $\alpha ,\beta \in \mathbb {F}$, and $[L^*]=[L]^*$.

(c) Show that the composition of linear maps satisfies $[ML]=[M][L]$.

(d) Suppose $L:{\mathscr {H}}\rightarrow {\mathscr {H}}$, and $W=V$. Show that f is an eigenvector of L for the eigenvalue $\lambda$ if and only if $Ax=\lambda x$, i.e., eigenvectors of L correspond to the eigenvectors of A that are in the range of $V^*$.

(e) Show L and A have the same singular values, and hence the same rank.