Notation
Wotao Yin
July 1, 2013
• real: R; complex: C; integer: Z.
• Dimensions: vector/signal N or n, linear measurements M or m, sparsity K or k.
• Sample index set: Ω.
• Point set: S, T , E, ...
• Scalars: a, b, d . . . (c is reserved for constants or certain parameter)
• Vectors: a, b, . . . and they are column vectors. Vector n is reserved for noise.
• 0 = [0, 0, . . . , 0]T and 1 = [1, 1, . . . , 1]T are column vectors
• sign(x) ∈ {−1, 0, 1} is the sign of x. sign(x) = [sign(x1 ), . . . , sign(xN )]T .
• Vector: x = [x1 , x2 , . . . , xN ]T with index starting from 1; component xi .
N ×N
• Sparsifying basis or dictionary: D = {di }N
, sparse coefficients s = [s1 , s2 , . . . , sN ]T .
i=1 ∈ C
x=
N
X
si di ,
and x = Ds.
i=1
• Measurements: b = [b1 , b2 , . . . , bM ]T
• Sensing matrix: Φ ∈ CM ×N or A ∈ CM ×N (can also be real valued).
• General matrices: A, B, . . .. Use “M -by-N matrix A”, “A ∈ RM ×N ”, or “A ∈ CM ×N ” to specify size.
• The set of N -by-N symmetric matrices: SN ; symmetric positive semi-definite: SN
+ ; symmetric positive
definite: SN
++ .
• The transpose of a matrix A is AT . The conjugate transpose is A∗ .
• RIP constants δK , δ2K , . . .
• Reserve c, c1 , c2 , . . . and C, C1 , C2 , . . . for constants
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• Identity matrix: I
• Original (unknown) vector: xo , “o ” stands for original
• Estimate (solution) vector: x̂, the hat stands for “estimated”. For non-estimation optimization problems, use xopt or x∗ for its optimal solution.
• The `p -norm minimization problem (use `p not lp ):
min kxkp ,
x
s.t. Θx = y.
Above “s.t.” stands for “subject to” not “such that.”
• The estimated error size is σ, which is commonly used for deviation. For example, the BPDN problem
is
min kxkp ,
x
s.t. kΘx − yk2 ≤ σ
Another form:
min kxkp +
x
µ
kΘx − yk22 .
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• Residual r := b − Ax.
• We write an M -by-N matrix A as


aT1
 . 
 . 
 . 
   
A =  aTi  = a1 , · · · , aj , · · · , aN = [aij ]1≤i≤M,1≤j≤N .
 
 . 
 .. 
 
aTM
Hence, the ith row of A is aTi , where ai is a column vector.
• support of x: supp(x);
• probability: Prob(· · · );
• diagonal matrix generated from vector x: Diag(x); vector extracted from the diagonal entries of A:
diag(A);
• domain of function f : dom(f );
• gradient g of function f at x: g(x) = ∇f (x);
• Hessian of function f at x: H(x) = ∇2 f (x);
• trace of matrix A: tr(A);
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• projection of point x to set S: ProjS (x)
• see macros.tex for other definitions of symbols and operators
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