multilinear rank of a tensor
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let $A$ be a tensor in $R^{n_1times...times n_d}$
if we use the truncated higher order singular value decomposition THOSVD to approximate this tensor to a tensor $B$ of multilinear rank $(s_1,...,s_d)$,
then $B=(U_1,...,U_d)C$ when $U_iin R^{n_itimes s_i}$ contain the first $s_i$ left singular vector of the $ith$ matricization of $A$ denoted by $A_{(i)}$ and $C=(U_1^T,...,U_d^T)A$
my question is : how to proove that $B_{(i)}$ is of rank $s_i$
tensors multilinear-algebra
asked Nov 15 at 10:10
Rima Khouja
965
add a comment |
up vote
0
down vote
favorite
let $A$ be a tensor in $R^{n_1times...times n_d}$
if we use the truncated higher order singular value decomposition THOSVD to approximate this tensor to a tensor $B$ of multilinear rank $(s_1,...,s_d)$,
then $B=(U_1,...,U_d)C$ when $U_iin R^{n_itimes s_i}$ contain the first $s_i$ left singular vector of the $ith$ matricization of $A$ denoted by $A_{(i)}$ and $C=(U_1^T,...,U_d^T)A$
my question is : how to proove that $B_{(i)}$ is of rank $s_i$
tensors multilinear-algebra
asked Nov 15 at 10:10
Rima Khouja
965
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
let $A$ be a tensor in $R^{n_1times...times n_d}$
if we use the truncated higher order singular value decomposition THOSVD to approximate this tensor to a tensor $B$ of multilinear rank $(s_1,...,s_d)$,
then $B=(U_1,...,U_d)C$ when $U_iin R^{n_itimes s_i}$ contain the first $s_i$ left singular vector of the $ith$ matricization of $A$ denoted by $A_{(i)}$ and $C=(U_1^T,...,U_d^T)A$
my question is : how to proove that $B_{(i)}$ is of rank $s_i$
tensors multilinear-algebra
asked Nov 15 at 10:10
Rima Khouja
965
let $A$ be a tensor in $R^{n_1times...times n_d}$
if we use the truncated higher order singular value decomposition THOSVD to approximate this tensor to a tensor $B$ of multilinear rank $(s_1,...,s_d)$,
then $B=(U_1,...,U_d)C$ when $U_iin R^{n_itimes s_i}$ contain the first $s_i$ left singular vector of the $ith$ matricization of $A$ denoted by $A_{(i)}$ and $C=(U_1^T,...,U_d^T)A$
my question is : how to proove that $B_{(i)}$ is of rank $s_i$
tensors multilinear-algebra
tensors multilinear-algebra
asked Nov 15 at 10:10
Rima Khouja
965
asked Nov 15 at 10:10
Rima Khouja
965
asked Nov 15 at 10:10
Rima Khouja
965
asked Nov 15 at 10:10
Rima Khouja
965
asked Nov 15 at 10:10
Rima Khouja
965
965
add a comment |
add a comment |
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