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 \((f_j)_{j\in J}\).

\({\mathscr {H}}\) – a real or complex Hilbert space (for us usually finite dimensional)

J – an index set (often with a group structure)

\((f_j)_{j\in J}\) – a sequence (set, or multiset) of vectors in \({\mathscr {H}}\)

\(\sum _{j\in J}\) – a sum (for us usually finite, but sometimes continuous)
The emphasis here is on the possible redundancy (over completeness) of the vectors \((f_j)\) in the expansion, i.e., the case when (2.1) is not an orthogonal expansion.
In the first instance, you are encouraged to consider \({\mathscr {H}}\) as \(\mathbb {R}^d\) or \(\mathbb {C}^d\), with the usual (Euclidean) inner product, and to think in familiar matrix terms.
2.1 Normalised tight frames
Definition 2.1.
For \(A{=}1\) (2.2) says that \(f\mapsto (\langle f, f_j\rangle )_{j\in J}\) is an isometry, and so normalised tight frames for \({\mathscr {H}}\) are equivalent to isometries \({\mathscr {H}}\rightarrow \ell _2(J)\). The maps taking normalised tight frames to normalised tight frames are the partial isometries (Exer. 2.7).
We prefer the term normalised tight frame ^{1} to Parseval frame (which is also used), as it emphasizes the fact the frame bound \(A>0\) is simply a normalising factor, i.e., if \((f_j)\) is a tight frame, then \((f_j/\sqrt{A})\) is the unique positive scalar multiple of it which is a normalised tight frame. We will soon see that this normalised version of a tight frame is convenient in many situations.
We have defined a tight frame to be a sequence, which is standard, but not universal. By contrast with a basis (which can be a set or a sequence), a tight frame can have repeated vectors. At times, e.g., when the vectors in a frame are all distinct or the indexing is unimportant, it can be convenient to think of them as a (multi)set. We will not labour this point, making statements such as the set \(\{f_j\}_{j\in J}\) is a tight frame, without further explanation.
Example 2.1.
(Exer. 2.4) An orthonormal basis is a normalised tight frame. These are the only normalised tight frames in which all the vectors have unit length (all vectors in a normalised tight frame have length \(\le 1\)).
Example 2.2.
(Exer. 2.5) The unitary image of a normalised tight frame is again a normalised tight frame, and the only invertible linear maps which map a normalised tight frame to a normalised tight frame are the unitary maps.
Example 2.3.
(Exer. 2.6) The orthogonal projection of a normalised tight frame is again a normalised tight frame (for its span). In particular, if U is an \(n\times n\) unitary matrix, then the columns of any \(d\times n\) submatrix is a normalised tight frame of n vectors for \(\mathbb {C}^d\). This is effectively the projection of the orthonormal basis for \(\mathbb {C}^n\) given by the columns of U onto the ddimensional subspace of vectors which are zero in some fixed \(nd\) coordinates.
We say that \((f_j)_{j\in J}\) is an equalnorm tight frame if \(\Vert f_j\Vert =\Vert f_k\Vert \), \(\forall j, k\in J\), and is a unitnorm tight frame if \(\Vert f_j\Vert =1\), \(\forall j\in J\).
Example 2.4.
In §2.6, we will show that every normalised tight frame can be obtained as the orthogonal projection of an orthonormal basis (in a larger space).
Example 2.5.
(Exer. 2.9) The tight frames for \(\mathbb {R}^2\). A sequence of vectors \((v_j)_{j=1}^n\), \(v_j=(x_j, y_j)\in \mathbb {R}^2\) is a tight frame for \(\mathbb {R}^2\) if and only if the diagram vectors which are defined by \(w_j := (x_j+i y_j)^2\in \mathbb {C}\) , \(1\le j\le n\), sum to zero (in \(\mathbb {C}\)).
2.2 Unitarily equivalent finite tight frames
Before giving any further concrete examples of finite tight frames, we define an equivalence, under which any set of three equally spaced vectors with the same norm in \(\mathbb {R}^2\) would be considered equivalent.
Definition 2.2.
We say that two normalised tight frames \((f_j)_{j\in J}\) for \({\mathscr {H}}\) and \((g_j)_{j\in J}\) for \({\mathscr {K}}\), with the same index set J, are (unitarily) equivalent if there is a unitary transformation \(U:{\mathscr {H}}\rightarrow {\mathscr {K}}\), such that \(g_j=Uf_j\) , \(\forall j\in J\).
Since unitary transformations preserve inner products, unitarily equivalent tight frames have the same inner products (angles) between their vectors. Furthermore, these inner products uniquely determine the equivalence classes (see §2.5).
This equivalence is dependent on the indexing, which is appropriate when set J has some natural (e.g., group) structure. The normalised tight frames of two vectors \((e_1,0)\) and \((0,e_1)\) for the onedimensional space \({\mathscr {H}}=\mathop {\mathrm{span}}\nolimits \{e_1\}\) are not equivalent, since there is no unitary map \(e_1\mapsto 0\) (or \(0\mapsto e_1\)). For such cases, where it is useful to consider these as equivalent, we extend our definition of equivalence as follows.
Definition 2.3.
We say that two finite normalised tight frames \((f_j)_{j\in J}\) for \({\mathscr {H}}\) and \((g_j)_{j\in K}\) for \({\mathscr {K}}\) are (unitarily) equivalent up to reordering if there is a bijection \(\sigma :J\rightarrow K\) for which \((f_j)_{j\in J}\) and \((g_{\sigma j})_{j\in J}\) are unitarily equivalent.
Example 2.6.
2.3 Projective and complex conjugate equivalences
All tight frames \((\alpha _j f_j)\), \(\alpha _j=1\), \(\forall j\), obtained from a given tight frame \((f_j)\), are projectively unitarily equivalent, but are not unitarily equivalent, in general.
Example 2.7.
For tight frames of n nonzero vectors in \(\mathbb {R}^2\) the equivalence classes for projective equivalence up to reordering are in 1–1 correspondence with convex polygons with n sides (see Exer. 2.10).
2.4 The analysis, synthesis and frame operators
For simplicity of presentation, we suppose J is finite, write \(\mathbb {F}\) for \(\mathbb {R}\) or \(\mathbb {C}\), \(\ell _2(J)\) for \(\mathbb {F}^J\), with the usual inner product, and \(I=I_{\mathscr {H}}\) for the identity on \({\mathscr {H}}\).
Definition 2.4.
It is convenient to make little distinction between the sequence \((f_j)_{j\in J}\) and the linear map \(V=[f_j]_{j\in J}\), which we will say has j th column \(f_j\).
Proposition 2.1.
Proof.
The equations (2.3) and (2.8) will be referred to as the Parseval identity, (2.9) as the trace formula, and (2.10) as the variational formula.
For \({\mathscr {H}}=\mathbb {F}^d\) and \(J=n\), V is a \(d\times n\) matrix, and the condition (2.8) says that the columns of V are orthogonal and of length \(\sqrt{A}\), i.e., \(V/\sqrt{A}\) is a coisometry, equivalently, \(V^*/\sqrt{A}\) is an isometry.
In § 6.2, we show that the variational formula characterises tight frames for finite dimensional spaces. There is no infinite dimensional counterpart for this result.
2.5 The Gramian
Unitary equivalence has the advantage (over projective unitary equivalence) that it preserves the inner product between vectors, and hence the Gramian matrix. Indeed, we will show that the Gramian characterises the equivalence class.
Definition 2.5.
This is the matrix representing the linear map \(V^*V:\ell _2(J)\rightarrow \ell _2(J)\) with respect to the standard orthonormal basis \(\{e_j\}_{j\in J}\).
The possible Gramian matrices are precisely the orthogonal projections:
Theorem 2.1.
Proof.
Finally, taking the trace of P gives (2.11). \(\square \)
The condition that \(P=\mathop {\mathrm{Gram}}\nolimits (\varPhi )\) be an orthogonal projection is equivalent to it having exactly d nonzero eigenvalues all equal to 1 (see Exer. 2.17).
Corollary 2.1.
(Characterisation of unitary equivalence) Normalised tight frames are unitarily equivalent if and only if their Gramians are equal.
Proof.
Let \(\varPhi =(f_j)_{j\in J}\), \(\varPsi =(g_j)_{j\in J}\) be normalised tight frames for \({\mathscr {H}}\) and \({\mathscr {K}}\).
In other words:
The properties of a tight frame (up to unitary equivalence) are determined by its Gramian.
Example 2.8.
2.6 Tight frames as orthogonal projections
We have seen (Exer. 2.6) that the orthogonal projection of an orthonormal basis is a normalised tight frame (for its span). The converse is also true.
Theorem 2.2.
Proof.
\(\square \)
When \(\varPsi \) and \(\varPhi \) are unitarily equivalent, then we will say that \(\varPsi \) is a copy of \(\varPhi \). With this terminology, Naĭmark’s theorem says:
A canonical copy of a tight frame \(\varPhi \) is given by the columns of \(\mathop {\mathrm{Gram}}\nolimits (\varPhi )\).
This is one of those often rediscovered theorems, which can be considered as a special case of Naĭmark’s theorem (see [AG63] and Exer. 2.26). Hadwiger [Had40] showed that \((f_j)_{j=1}^n\) in \(\mathbb {R}^d\) is a coordinate star (normalised tight frame) if and only if it is a Pohlke normal star (projection of an orthonormal basis). In signal processing, this method of obtaining tight frames is called seeding.
Example 2.9.
Example 2.10.
A cross in \(\mathbb {R}^n\) is the set obtained by taking an orthonormal basis and its negatives \(\{\pm e_1,\ldots ,\pm e_n\}\), and the orthogonal projection of a cross onto a ddimensional subspace V is a called a eutactic star (see Coxeter [Cox73]). In view of Theorem 2.2, a eutactic star is precisely a tight frame of the form \(\{\pm a_1,\ldots ,\pm a_n\}\) for V, i.e., the union of a tight frame \(\{a_1,\ldots , a_n\}\) and the equivalent frame obtained by taking its negative. When the vectors \(a_i\) all have the same length, one obtains a socalled normalised eutactic star. Since equalnorm tight frames always exist (see Chapters 7 and 11), so do normalised eutactic stars in \(\mathbb {R}^d\) for every \(n\ge d\).
2.7 The construction of tight frames from orthogonal projections
The Gramian of a normalised tight frame \(\varPhi =(v_j)_{j=1}^n\) for a ddimensional space is an orthogonal projection P. By Theorem 2.2, the columns \((Pe_j)\) of P give a canonical copy of the frame (up to unitary equivalence) as a ddimensional subspace of \(\mathbb {F}^n\). To obtain a copy of \(\varPhi \) in \(\mathbb {F}^d\), we consider the rows of \(P=\mathop {\mathrm{Gram}}\nolimits (\varPhi )\).
Theorem 2.3.
(Row construction). Let \(P\in \mathbb {C}^{n\times n}\) be an orthogonal projection matrix of rank d. The columns of \(V=[v_1,\ldots , v_n]\in \mathbb {C}^{d\times n}\) are a normalised tight frame for \(\mathbb {C}^d\) with Gramian P if and only if the rows of V are an orthonormal basis for the row space of P. In particular, such a V can always be obtained by applying the Gram–Schmidt process to the rows of P.
Proof.
In other words:
A frame \(V=[v_1,\ldots , v_n]\) is a copy of a normalised tight frame \(\varPhi \) if and only if the rows of V are an orthonormal basis for the row space of \(\mathop {\mathrm{Gram}}\nolimits (\varPhi )\).
Example 2.11.
Example 2.12.
2.8 Complementary tight frames
Tight frames are determined (up to unitary equivalence) by their Gramian matrix P, which is an orthogonal projection matrix (when the frame is normalised), and all orthogonal projection matrices correspond to normalised tight frames. Thus there is normalised tight frame with Gramian given by the complementary projection \(IP\).
Definition 2.6.
Given a finite normalised tight frame \(\varPhi \) with Gramian P, we call any normalised tight frame with Gramian \(IP\) its complement. More generally, we say that two tight frames are complements of each other, if after normalisation the sum of their Gramians is the identity I.
In view of (2.11), the complement of a tight frame of n vectors for a space of dimension d is a tight frame of n vectors for a space of dimension \(nd\).
A tight frame and its complement can never be unitarily or projectively unitarily equivalent (Exer. 2.23), though they do have the same symmetries (see § 9.2, § 9.3).
Example 2.13.
The complement of an orthonormal basis for \(\mathbb {C}^d\) is the frame for the zero vector space given by d zero vectors.
Example 2.14.
2.9 Partition frames
Definition 2.7.
Example 2.15.
(Simplex) For \(n=d+1\), the trivial partition \(\alpha =(d+1)\) gives the vertices of the simplex in \(\mathbb {R}^d\).
Partition  n  Description of partition frame  G 

(3)  3  three equally spaced vectors in \(\mathbb {R}^2\)  6 
(2, 2)  4  four equally spaced vectors in \(\mathbb {R}^2\)  8 
(4)  4  vertices of the tetrahedron in \(\mathbb {R}^3\)  24 
(2, 3)  5  vertices of the trigonal bipyramid in \(\mathbb {R}^3\)  12 
(2, 2, 2)  6  vertices of the octahedron in \(\mathbb {R}^3\)  48 
2.10 Real and complex tight frames
A tight frame for a real Hilbert space is a tight frame for its complexification (see Exer. 2.28). We will call frames that come in this way real tight frames.
Definition 2.8.
We say that a tight frame \((f_j)_{j\in J}\) is real if its Gramian is a real matrix, and otherwise it is complex.
By Theorem 2.3 (row construction), a tight frame for a space \({\mathscr {H}}\) of dimension d is real if and only if there is a unitary matrix \(U:{\mathscr {H}}\rightarrow \mathbb {F}^d\) for which \(Uf_j\in \mathbb {R}^d\), \(\forall j\). Moreover, a frame is complex if and only if its complementary frame is.
Example 2.16.
There are intrinsic differences between the classes of real and complex frames, e.g., see Exer. 1.3, § 6.8, § 12.1, § 12.10 and § 12.17.
The real algebraic variety of real and complex normalised finite tight frames (and unitnorm tight frames) is considered in Chapter 7.
Remark 2.1.
One could extend the Definition 2.8 to other fields, e.g., say that the three equally spaced unit vectors in \(\mathbb {R}^2\) are a rational tight frame, since their Gramian has rational entries. In this case, the columns of the Gramian give a copy of this frame in a rational inner product space (Example 2.9), but the row construction (Example 2.11) does not give a copy in \(\mathbb {Q}^2\) (with the Euclidean inner product). These ideas are explored in [CFW15].
2.11 SICs and MUBs
There are many interesting and useful examples of equalnorm tight frames \((v_j)\), e.g., group frames (see § 10). Those for which the crosscorrelation \(\langle v_j, v_k\rangle \), \(j\ne k\), takes a small number of values are of particular interest (especially for applications). We briefly mention two such classes of frames: the SICs and the MUBs. These are simple to describe, and come with some intriguing conjectures, which are still unproven (despite considerable work on them). Indeed, the construction of SICs (see § 14) and maximal sets of MUBs (see § 14.2, § 8.6) are two central problems in the theory of finite tight frames.
Definition 2.9.
SICs can be viewed as maximal sets of complex equiangular lines. It follows from the bounds of Theorem 12.2 on such lines that SICs are complex frames. Their origins as quantum measurements and the known constructions are detailed in Chapter 14. The conjecture that SICs exist in every dimension d is known as Zauner’s conjecture or the SIC problem.
Example 2.17.
The second prototypical example ( 1.7) is a SIC for \(\mathbb {C}^2\). The case \(d=3\) seems to be an exception for SICs. Here the SICs form a continuous family, while for \(d\ne 3\), there is currently only a finite number of SICs for \(\mathbb {C}^d\) known.
Definition 2.10.
Example 2.18.
Footnotes
 1.
This term dates back to [HL00]. Just to confuse matters, the term normalised tight frame has also been used for a tight frame with \(\Vert f_j\Vert =1\), \(\forall j\in J\) (we call these unitnorm tight frames).
 2.
Note the (j, k)entry of the Gramian is \(\langle f_k, f_j\rangle =f_j^*f_k\) (so it factors \(V^*V\)), not \(\langle f_j, f_k\rangle \), which is sometimes used to define the Gramian.