Mixing
Calculating the marginal expectation of a joint pdf on the unit sphere
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Consider a random matrix $mathbfX in mathbbC^Ttimes M$ with independent columns $mathbfx_1,dots,mathbfx_M$ distributed on the unit sphere $mathcalS = = 1$. That is, $|mathbfx_i| = 1$, for $i = 1,dots,M$. The joint probability density function of the columns is given by
beginalign
f(mathbfX) &= frac1Q fracexp(trmathbfY^H(sigma^2mathbfI_T + mathbfXmathbfX^H)^-1mathbfY+trmathbfZ^HmathbfX)det(sigma^2I_T + mathbfXmathbfX^H),
endalign
where $mathbfY in mathbbC^Ttimes N$ and $mathbfZ = [mathbfz_1 dots mathbfz_M] in mathbbC^Ttimes M$ are deterministic matrices, $sigma^2 < 1$, $$Q = int_mathcalS^Mfracexp(trmathbfY^H(sigma^2mathbfI_T + mathbfXmathbfX^H)^-1mathbfY+trmathbfZ^HmathbfX)det(sigma^2I_T + mathbfXmathbfX^H)dmathbfX$$ is the normalization factor. In other words,
beginalign
f(mathbfX) = frac1Qfracexp(trmathbfY^H(sigma^2mathbfI_T + mathbfXmathbfX^H)^-1mathbfY)det(sigma^2I_T + mathbfXmathbfX^H)prod_i = 1^M exp(mathbfz_i^Hmathbfx_i).
endalign
I want to derive the marginal expectation $mathbbE[mathbfx_i] = int_mathcalS^M f(mathbfX)mathbfx_i dmathbfX$, $i = 1,dots,M$.
This can be solved by factorizing $f(mathbfX)$ as a product of functions of each column of $mathbfX$. However, the term $fracexp(trmathbfY^H(sigma^2mathbfI_T + mathbfXmathbfX^H)^-1mathbfY)det(sigma^2I_T + mathbfXmathbfX^H)$ makes it difficult and I understand that it is not hopeful for an exact expession of $mathbbE[mathbfx_i]$. So, I am looking for a reasonable approximation of $f(mathbfX)$ which can be marginalized more easily.
A straightforward approximation is to impose that the columns of $mathbfX$ are orthogonal, then $mathbfX^HmathbfX = mathbfI_M$ and we get rid of $fracexp(trmathbfY^H(sigma^2mathbfI_T + mathbfXmathbfX^H)^-1mathbfY)det(sigma^2I_T + mathbfXmathbfX^H)$. However, this approximation is very loose.
Do you have any idea for a sharper marginalizable approximation?
Thank you.
expectation approximation
asked Aug 2 at 12:05
Khac-Hoang Ngo
113
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up vote
1
down vote
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Consider a random matrix $mathbfX in mathbbC^Ttimes M$ with independent columns $mathbfx_1,dots,mathbfx_M$ distributed on the unit sphere $mathcalS = = 1$. That is, $|mathbfx_i| = 1$, for $i = 1,dots,M$. The joint probability density function of the columns is given by
beginalign
f(mathbfX) &= frac1Q fracexp(trmathbfY^H(sigma^2mathbfI_T + mathbfXmathbfX^H)^-1mathbfY+trmathbfZ^HmathbfX)det(sigma^2I_T + mathbfXmathbfX^H),
endalign
where $mathbfY in mathbbC^Ttimes N$ and $mathbfZ = [mathbfz_1 dots mathbfz_M] in mathbbC^Ttimes M$ are deterministic matrices, $sigma^2 < 1$, $$Q = int_mathcalS^Mfracexp(trmathbfY^H(sigma^2mathbfI_T + mathbfXmathbfX^H)^-1mathbfY+trmathbfZ^HmathbfX)det(sigma^2I_T + mathbfXmathbfX^H)dmathbfX$$ is the normalization factor. In other words,
beginalign
f(mathbfX) = frac1Qfracexp(trmathbfY^H(sigma^2mathbfI_T + mathbfXmathbfX^H)^-1mathbfY)det(sigma^2I_T + mathbfXmathbfX^H)prod_i = 1^M exp(mathbfz_i^Hmathbfx_i).
endalign
I want to derive the marginal expectation $mathbbE[mathbfx_i] = int_mathcalS^M f(mathbfX)mathbfx_i dmathbfX$, $i = 1,dots,M$.
This can be solved by factorizing $f(mathbfX)$ as a product of functions of each column of $mathbfX$. However, the term $fracexp(trmathbfY^H(sigma^2mathbfI_T + mathbfXmathbfX^H)^-1mathbfY)det(sigma^2I_T + mathbfXmathbfX^H)$ makes it difficult and I understand that it is not hopeful for an exact expession of $mathbbE[mathbfx_i]$. So, I am looking for a reasonable approximation of $f(mathbfX)$ which can be marginalized more easily.
A straightforward approximation is to impose that the columns of $mathbfX$ are orthogonal, then $mathbfX^HmathbfX = mathbfI_M$ and we get rid of $fracexp(trmathbfY^H(sigma^2mathbfI_T + mathbfXmathbfX^H)^-1mathbfY)det(sigma^2I_T + mathbfXmathbfX^H)$. However, this approximation is very loose.
Do you have any idea for a sharper marginalizable approximation?
Thank you.
expectation approximation
asked Aug 2 at 12:05
Khac-Hoang Ngo
113
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
Consider a random matrix $mathbfX in mathbbC^Ttimes M$ with independent columns $mathbfx_1,dots,mathbfx_M$ distributed on the unit sphere $mathcalS = = 1$. That is, $|mathbfx_i| = 1$, for $i = 1,dots,M$. The joint probability density function of the columns is given by
beginalign
f(mathbfX) &= frac1Q fracexp(trmathbfY^H(sigma^2mathbfI_T + mathbfXmathbfX^H)^-1mathbfY+trmathbfZ^HmathbfX)det(sigma^2I_T + mathbfXmathbfX^H),
endalign
where $mathbfY in mathbbC^Ttimes N$ and $mathbfZ = [mathbfz_1 dots mathbfz_M] in mathbbC^Ttimes M$ are deterministic matrices, $sigma^2 < 1$, $$Q = int_mathcalS^Mfracexp(trmathbfY^H(sigma^2mathbfI_T + mathbfXmathbfX^H)^-1mathbfY+trmathbfZ^HmathbfX)det(sigma^2I_T + mathbfXmathbfX^H)dmathbfX$$ is the normalization factor. In other words,
beginalign
f(mathbfX) = frac1Qfracexp(trmathbfY^H(sigma^2mathbfI_T + mathbfXmathbfX^H)^-1mathbfY)det(sigma^2I_T + mathbfXmathbfX^H)prod_i = 1^M exp(mathbfz_i^Hmathbfx_i).
endalign
I want to derive the marginal expectation $mathbbE[mathbfx_i] = int_mathcalS^M f(mathbfX)mathbfx_i dmathbfX$, $i = 1,dots,M$.
This can be solved by factorizing $f(mathbfX)$ as a product of functions of each column of $mathbfX$. However, the term $fracexp(trmathbfY^H(sigma^2mathbfI_T + mathbfXmathbfX^H)^-1mathbfY)det(sigma^2I_T + mathbfXmathbfX^H)$ makes it difficult and I understand that it is not hopeful for an exact expession of $mathbbE[mathbfx_i]$. So, I am looking for a reasonable approximation of $f(mathbfX)$ which can be marginalized more easily.
A straightforward approximation is to impose that the columns of $mathbfX$ are orthogonal, then $mathbfX^HmathbfX = mathbfI_M$ and we get rid of $fracexp(trmathbfY^H(sigma^2mathbfI_T + mathbfXmathbfX^H)^-1mathbfY)det(sigma^2I_T + mathbfXmathbfX^H)$. However, this approximation is very loose.
Do you have any idea for a sharper marginalizable approximation?
Thank you.
expectation approximation
asked Aug 2 at 12:05
Khac-Hoang Ngo
113
Consider a random matrix $mathbfX in mathbbC^Ttimes M$ with independent columns $mathbfx_1,dots,mathbfx_M$ distributed on the unit sphere $mathcalS = = 1$. That is, $|mathbfx_i| = 1$, for $i = 1,dots,M$. The joint probability density function of the columns is given by
beginalign
f(mathbfX) &= frac1Q fracexp(trmathbfY^H(sigma^2mathbfI_T + mathbfXmathbfX^H)^-1mathbfY+trmathbfZ^HmathbfX)det(sigma^2I_T + mathbfXmathbfX^H),
endalign
where $mathbfY in mathbbC^Ttimes N$ and $mathbfZ = [mathbfz_1 dots mathbfz_M] in mathbbC^Ttimes M$ are deterministic matrices, $sigma^2 < 1$, $$Q = int_mathcalS^Mfracexp(trmathbfY^H(sigma^2mathbfI_T + mathbfXmathbfX^H)^-1mathbfY+trmathbfZ^HmathbfX)det(sigma^2I_T + mathbfXmathbfX^H)dmathbfX$$ is the normalization factor. In other words,
beginalign
f(mathbfX) = frac1Qfracexp(trmathbfY^H(sigma^2mathbfI_T + mathbfXmathbfX^H)^-1mathbfY)det(sigma^2I_T + mathbfXmathbfX^H)prod_i = 1^M exp(mathbfz_i^Hmathbfx_i).
endalign
I want to derive the marginal expectation $mathbbE[mathbfx_i] = int_mathcalS^M f(mathbfX)mathbfx_i dmathbfX$, $i = 1,dots,M$.
This can be solved by factorizing $f(mathbfX)$ as a product of functions of each column of $mathbfX$. However, the term $fracexp(trmathbfY^H(sigma^2mathbfI_T + mathbfXmathbfX^H)^-1mathbfY)det(sigma^2I_T + mathbfXmathbfX^H)$ makes it difficult and I understand that it is not hopeful for an exact expession of $mathbbE[mathbfx_i]$. So, I am looking for a reasonable approximation of $f(mathbfX)$ which can be marginalized more easily.
A straightforward approximation is to impose that the columns of $mathbfX$ are orthogonal, then $mathbfX^HmathbfX = mathbfI_M$ and we get rid of $fracexp(trmathbfY^H(sigma^2mathbfI_T + mathbfXmathbfX^H)^-1mathbfY)det(sigma^2I_T + mathbfXmathbfX^H)$. However, this approximation is very loose.
Do you have any idea for a sharper marginalizable approximation?
Thank you.
expectation approximation
asked Aug 2 at 12:05
Khac-Hoang Ngo
113
edited Aug 2 at 12:52
asked Aug 2 at 12:05
Khac-Hoang Ngo
113
asked Aug 2 at 12:05
Khac-Hoang Ngo
113
asked Aug 2 at 12:05
Khac-Hoang Ngo
113
113
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