# on 09-Jan-2018 (Tue)

#### Annotation 1729520602380

 #matrix-inversion For , a pseudoinverse of is defined as a matrix satisfying all of the following four criteria: ( AA+ need not be the general identity matrix, but it maps all column vectors of A to themselves); ( A+ is a weak inverse for the multiplicative semigroup); ( AA+ is Hermitian); and ( A+A is also Hermitian). Moore-Penrose Pseudo-inverse exists for any matrix , but when the latter has full rank, can be expressed as a simple algebraic formula. In particular, when has linearly independent columns (and thus matrix is invertible), can be computed as: ...

Moore–Penrose inverse - Wikipedia
; K ) {\displaystyle I_{n}\in \mathrm {M} (n,n;K)} denotes the n × n {\displaystyle n\times n} identity matrix. Definition[edit source] <span>For A ∈ M ( m , n ; K ) {\displaystyle A\in \mathrm {M} (m,n;K)} , a pseudoinverse of A {\displaystyle A} is defined as a matrix A + ∈ M ( n , m ; K ) {\displaystyle A^{+}\in \mathrm {M} (n,m;K)} satisfying all of the following four criteria: [8] [9] A A + A = A {\displaystyle AA^{+}A=A\,\!} (AA + need not be the general identity matrix, but it maps all column vectors of A to themselves); A + A A + = A + {\displaystyle A^{+}AA^{+}=A^{+}\,\!} (A + is a weak inverse for the multiplicative semigroup); ( A A + ) ∗ = A A + {\displaystyle (AA^{+})^{*}=AA^{+}\,\!} (AA + is Hermitian); and ( A + A ) ∗ = A + A {\displaystyle (A^{+}A)^{*}=A^{+}A\,\!} (A + A is also Hermitian). A + {\displaystyle A^{+}} exists for any matrix A {\displaystyle A} , but when the latter has full rank, A + {\displaystyle A^{+}} can be expressed as a simple algebraic formula. In particular, when A {\displaystyle A} has linearly independent columns (and thus matrix A ∗ A {\displaystyle A^{*}A} is invertible), A + {\displaystyle A^{+}} can be computed as: A + = ( A ∗ A ) − 1 A ∗ . {\displaystyle A^{+}=(A^{*}A)^{-1}A^{*}\,.} This particular pseudoinverse constitutes a left inverse, since, in this case, A + A = I {\displaystyle A^{+}A=I} . When A {\displaystyle A} has linearly independent rows (matrix A A ∗ {\displaystyle AA^{*}} is invertible), A + {\displaystyle A^{+}} can be computed as: A + = A ∗ ( A A ∗ ) − 1 . {\displaystyle A^{+}=A^{*}(AA^{*})^{-1}\,.} This is a right inverse, as A A + = I {\displaystyle AA^{+}=I} . Properties[edit source] Proofs for some of these facts may be found on a separate page Proofs involving the Moore–Penrose inverse. Existence and uniqueness[edit source] The pseu

#### Annotation 1731089534220

 pseudo datapoint based approximation methods for DGPs trade model complexity for a lower computational complexity of $$O(NLM^ 2 )$$ where N is the number of datapoints, L is the number of layers, and M is the number of pseudo datapoints. This complexity scales quadratically in M whereas the dependence on the number of layers L is only linear. Therefore, it can be cheaper to increase the representation power of the model by adding extra layers rather than by adding more pseudo datapoints

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#### Annotation 1731435564300

 #singular-value-decomposition SVD as change of coordinates The geometric content of the SVD theorem can thus be summarized as follows: for every linear map T : Kn → Km one can find orthonormal bases of Kn and Km such that T maps the i-th basis vector of Kn to a non-negative multiple of the i-th basis vector of Km , and sends the left-over basis vectors to zero. With respect to these bases, the map T is therefore represented by a diagonal matrix with non-negative real diagonal entries.

Singular-value decomposition - Wikipedia
m , n ) , {\displaystyle T(\mathbf {V} _{i})=\sigma _{i}\mathbf {U} _{i},\qquad i=1,\ldots ,\min(m,n),} where σ i is the i-th diagonal entry of Σ, and T(V i ) = 0 for i > min(m,n). <span>The geometric content of the SVD theorem can thus be summarized as follows: for every linear map T : K n → K m one can find orthonormal bases of K n and K m such that T maps the i-th basis vector of K n to a non-negative multiple of the i-th basis vector of K m , and sends the left-over basis vectors to zero. With respect to these bases, the map T is therefore represented by a diagonal matrix with non-negative real diagonal entries. To get a more visual flavour of singular values and SVD factorization — at least when working on real vector spaces — consider the sphere S of radius one in R n . The linear map T map

#### Annotation 1731438447884

 #matrix-inversion A computationally simple and accurate way to compute the pseudoinverse is by using the singular value decomposition.[1][9][15] If is the singular value decomposition of A , then . For a rectangular diagonal matrix such as Σ {\displaystyle \Sigma } , we get the pseudoinverse by taking the reciprocal of each non-zero element on the diagonal, leaving the zeros in place, and then transposing the matrix. In numerical computation, only elements larger than some small tolerance are taken to be nonzero, and the others are replaced by zeros. For example, in the MATLAB, GNU Octave, or NumPy function pinv , the tolerance is taken to be t = ε⋅max(m,n)⋅max(Σ) , where ε is the machine epsilon.

Moore–Penrose inverse - Wikipedia
A {\displaystyle A} and A ∗ {\displaystyle A^{*}} . Singular value decomposition (SVD)[edit source] <span>A computationally simple and accurate way to compute the pseudoinverse is by using the singular value decomposition. [1] [9] [15] If A = U Σ V ∗ {\displaystyle A=U\Sigma V^{*}} is the singular value decomposition of A, then A + = V Σ + U ∗ {\displaystyle A^{+}=V\Sigma ^{+}U^{*}} . For a rectangular diagonal matrix such as Σ {\displaystyle \Sigma } , we get the pseudoinverse by taking the reciprocal of each non-zero element on the diagonal, leaving the zeros in place, and then transposing the matrix. In numerical computation, only elements larger than some small tolerance are taken to be nonzero, and the others are replaced by zeros. For example, in the MATLAB, GNU Octave, or NumPy function pinv , the tolerance is taken to be t = ε⋅max(m,n)⋅max(Σ), where ε is the machine epsilon. The computational cost of this method is dominated by the cost of computing the SVD, which is several times higher than matrix–matrix multiplication, even if a state-of-the art implem

#### Annotation 1731441331468

 #matrix-inversion Moore-Penrose Pseudo-inverse exists for any matrix , but when the latter has full rank, can be expressed as a simple algebraic formula. In particular, when has linearly independent columns (and thus matrix is invertible), can be computed as: This particular pseudoinverse constitutes a left inverse, since, in this case, .

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3; ( AA + need not be the general identity matrix, but it maps all column vectors of A to themselves); ( A + is a weak inverse for the multiplicative semigroup); ( AA + is Hermitian); and ( A + A is also Hermitian). <span>Moore-Penrose Pseudo-inverse exists for any matrix , but when the latter has full rank, can be expressed as a simple algebraic formula. In particular, when has linearly independent columns (and thus matrix is invertible), can be computed as: This particular pseudoinverse constitutes a left inverse, since, in this case, . When has linearly independent rows (matrix is invertible), can be computed as: This is a right inverse, as . <span><body><html>

#### Original toplevel document

Moore–Penrose inverse - Wikipedia
; K ) {\displaystyle I_{n}\in \mathrm {M} (n,n;K)} denotes the n × n {\displaystyle n\times n} identity matrix. Definition[edit source] <span>For A ∈ M ( m , n ; K ) {\displaystyle A\in \mathrm {M} (m,n;K)} , a pseudoinverse of A {\displaystyle A} is defined as a matrix A + ∈ M ( n , m ; K ) {\displaystyle A^{+}\in \mathrm {M} (n,m;K)} satisfying all of the following four criteria: [8] [9] A A + A = A {\displaystyle AA^{+}A=A\,\!} (AA + need not be the general identity matrix, but it maps all column vectors of A to themselves); A + A A + = A + {\displaystyle A^{+}AA^{+}=A^{+}\,\!} (A + is a weak inverse for the multiplicative semigroup); ( A A + ) ∗ = A A + {\displaystyle (AA^{+})^{*}=AA^{+}\,\!} (AA + is Hermitian); and ( A + A ) ∗ = A + A {\displaystyle (A^{+}A)^{*}=A^{+}A\,\!} (A + A is also Hermitian). A + {\displaystyle A^{+}} exists for any matrix A {\displaystyle A} , but when the latter has full rank, A + {\displaystyle A^{+}} can be expressed as a simple algebraic formula. In particular, when A {\displaystyle A} has linearly independent columns (and thus matrix A ∗ A {\displaystyle A^{*}A} is invertible), A + {\displaystyle A^{+}} can be computed as: A + = ( A ∗ A ) − 1 A ∗ . {\displaystyle A^{+}=(A^{*}A)^{-1}A^{*}\,.} This particular pseudoinverse constitutes a left inverse, since, in this case, A + A = I {\displaystyle A^{+}A=I} . When A {\displaystyle A} has linearly independent rows (matrix A A ∗ {\displaystyle AA^{*}} is invertible), A + {\displaystyle A^{+}} can be computed as: A + = A ∗ ( A A ∗ ) − 1 . {\displaystyle A^{+}=A^{*}(AA^{*})^{-1}\,.} This is a right inverse, as A A + = I {\displaystyle AA^{+}=I} . Properties[edit source] Proofs for some of these facts may be found on a separate page Proofs involving the Moore–Penrose inverse. Existence and uniqueness[edit source] The pseu

#### Flashcard 1731444477196

Tags
#matrix-inversion
Question

when has [...] the Moore-Penrose inverse is a left inverse

linearly independent columns

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ml> Moore-Penrose Pseudo-inverse exists for any matrix , but when the latter has full rank, can be expressed as a simple algebraic formula. In particular, when has linearly independent columns (and thus matrix is invertible), can be computed as: This particular pseudoinverse constitutes a left inverse, since, in this case, . <html>

#### Original toplevel document

Moore–Penrose inverse - Wikipedia
; K ) {\displaystyle I_{n}\in \mathrm {M} (n,n;K)} denotes the n × n {\displaystyle n\times n} identity matrix. Definition[edit source] <span>For A ∈ M ( m , n ; K ) {\displaystyle A\in \mathrm {M} (m,n;K)} , a pseudoinverse of A {\displaystyle A} is defined as a matrix A + ∈ M ( n , m ; K ) {\displaystyle A^{+}\in \mathrm {M} (n,m;K)} satisfying all of the following four criteria: [8] [9] A A + A = A {\displaystyle AA^{+}A=A\,\!} (AA + need not be the general identity matrix, but it maps all column vectors of A to themselves); A + A A + = A + {\displaystyle A^{+}AA^{+}=A^{+}\,\!} (A + is a weak inverse for the multiplicative semigroup); ( A A + ) ∗ = A A + {\displaystyle (AA^{+})^{*}=AA^{+}\,\!} (AA + is Hermitian); and ( A + A ) ∗ = A + A {\displaystyle (A^{+}A)^{*}=A^{+}A\,\!} (A + A is also Hermitian). A + {\displaystyle A^{+}} exists for any matrix A {\displaystyle A} , but when the latter has full rank, A + {\displaystyle A^{+}} can be expressed as a simple algebraic formula. In particular, when A {\displaystyle A} has linearly independent columns (and thus matrix A ∗ A {\displaystyle A^{*}A} is invertible), A + {\displaystyle A^{+}} can be computed as: A + = ( A ∗ A ) − 1 A ∗ . {\displaystyle A^{+}=(A^{*}A)^{-1}A^{*}\,.} This particular pseudoinverse constitutes a left inverse, since, in this case, A + A = I {\displaystyle A^{+}A=I} . When A {\displaystyle A} has linearly independent rows (matrix A A ∗ {\displaystyle AA^{*}} is invertible), A + {\displaystyle A^{+}} can be computed as: A + = A ∗ ( A A ∗ ) − 1 . {\displaystyle A^{+}=A^{*}(AA^{*})^{-1}\,.} This is a right inverse, as A A + = I {\displaystyle AA^{+}=I} . Properties[edit source] Proofs for some of these facts may be found on a separate page Proofs involving the Moore–Penrose inverse. Existence and uniqueness[edit source] The pseu

#### Flashcard 1731448147212

Tags
#matrix-inversion
Question

The left Moore-Penrose Pseudo-inverse is [...]

This is the one for linear models

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e exists for any matrix , but when the latter has full rank, can be expressed as a simple algebraic formula. In particular, when has linearly independent columns (and thus matrix is invertible), can be computed as<span>: This particular pseudoinverse constitutes a left inverse, since, in this case, . <span><body><html>

#### Original toplevel document

Moore–Penrose inverse - Wikipedia
; K ) {\displaystyle I_{n}\in \mathrm {M} (n,n;K)} denotes the n × n {\displaystyle n\times n} identity matrix. Definition[edit source] <span>For A ∈ M ( m , n ; K ) {\displaystyle A\in \mathrm {M} (m,n;K)} , a pseudoinverse of A {\displaystyle A} is defined as a matrix A + ∈ M ( n , m ; K ) {\displaystyle A^{+}\in \mathrm {M} (n,m;K)} satisfying all of the following four criteria: [8] [9] A A + A = A {\displaystyle AA^{+}A=A\,\!} (AA + need not be the general identity matrix, but it maps all column vectors of A to themselves); A + A A + = A + {\displaystyle A^{+}AA^{+}=A^{+}\,\!} (A + is a weak inverse for the multiplicative semigroup); ( A A + ) ∗ = A A + {\displaystyle (AA^{+})^{*}=AA^{+}\,\!} (AA + is Hermitian); and ( A + A ) ∗ = A + A {\displaystyle (A^{+}A)^{*}=A^{+}A\,\!} (A + A is also Hermitian). A + {\displaystyle A^{+}} exists for any matrix A {\displaystyle A} , but when the latter has full rank, A + {\displaystyle A^{+}} can be expressed as a simple algebraic formula. In particular, when A {\displaystyle A} has linearly independent columns (and thus matrix A ∗ A {\displaystyle A^{*}A} is invertible), A + {\displaystyle A^{+}} can be computed as: A + = ( A ∗ A ) − 1 A ∗ . {\displaystyle A^{+}=(A^{*}A)^{-1}A^{*}\,.} This particular pseudoinverse constitutes a left inverse, since, in this case, A + A = I {\displaystyle A^{+}A=I} . When A {\displaystyle A} has linearly independent rows (matrix A A ∗ {\displaystyle AA^{*}} is invertible), A + {\displaystyle A^{+}} can be computed as: A + = A ∗ ( A A ∗ ) − 1 . {\displaystyle A^{+}=A^{*}(AA^{*})^{-1}\,.} This is a right inverse, as A A + = I {\displaystyle AA^{+}=I} . Properties[edit source] Proofs for some of these facts may be found on a separate page Proofs involving the Moore–Penrose inverse. Existence and uniqueness[edit source] The pseu

#### Flashcard 1731451292940

Tags
#singular-value-decomposition
Question
geometrically SVD finds [...] for every linear map T : KnKm
orthonormal bases of Kn and Km

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SVD as change of coordinates The geometric content of the SVD theorem can thus be summarized as follows: for every linear map T : K n → K m one can find orthonormal bases of K n and K m such that T maps the i-th basis vector of K n to a non-negative multiple of the i-th basis vector of K m , and sends the left-over basis vectors to zero. With respect to these bases,

#### Original toplevel document

Singular-value decomposition - Wikipedia
m , n ) , {\displaystyle T(\mathbf {V} _{i})=\sigma _{i}\mathbf {U} _{i},\qquad i=1,\ldots ,\min(m,n),} where σ i is the i-th diagonal entry of Σ, and T(V i ) = 0 for i > min(m,n). <span>The geometric content of the SVD theorem can thus be summarized as follows: for every linear map T : K n → K m one can find orthonormal bases of K n and K m such that T maps the i-th basis vector of K n to a non-negative multiple of the i-th basis vector of K m , and sends the left-over basis vectors to zero. With respect to these bases, the map T is therefore represented by a diagonal matrix with non-negative real diagonal entries. To get a more visual flavour of singular values and SVD factorization — at least when working on real vector spaces — consider the sphere S of radius one in R n . The linear map T map

#### Flashcard 1731453127948

Tags
#singular-value-decomposition
Question
Geometrically SVD finds orthonormal bases of Kn and Km for every linear map T : KnKm such that T maps the i-th basis vector of Kn to a non-negative multiple of the i-th basis vector of Km , and sends the left-over basis vectors to [...].
zero

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ollows: for every linear map T : K n → K m one can find orthonormal bases of K n and K m such that T maps the i-th basis vector of K n to a non-negative multiple of the i-th basis vector of K m , and sends the left-over basis vectors to <span>zero. With respect to these bases, the map T is therefore represented by a diagonal matrix with non-negative real diagonal entries. <span><body><html>

#### Original toplevel document

Singular-value decomposition - Wikipedia
m , n ) , {\displaystyle T(\mathbf {V} _{i})=\sigma _{i}\mathbf {U} _{i},\qquad i=1,\ldots ,\min(m,n),} where σ i is the i-th diagonal entry of Σ, and T(V i ) = 0 for i > min(m,n). <span>The geometric content of the SVD theorem can thus be summarized as follows: for every linear map T : K n → K m one can find orthonormal bases of K n and K m such that T maps the i-th basis vector of K n to a non-negative multiple of the i-th basis vector of K m , and sends the left-over basis vectors to zero. With respect to these bases, the map T is therefore represented by a diagonal matrix with non-negative real diagonal entries. To get a more visual flavour of singular values and SVD factorization — at least when working on real vector spaces — consider the sphere S of radius one in R n . The linear map T map

#### Flashcard 1731454700812

Tags
#singular-value-decomposition
Question
With SVD geometrically every linear map T : KnKm is represented by a diagonal matrix with [...] entries.
non-negative real diagonal

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h that T maps the i-th basis vector of K n to a non-negative multiple of the i-th basis vector of K m , and sends the left-over basis vectors to zero. With respect to these bases, the map T is therefore represented by a diagonal matrix with <span>non-negative real diagonal entries. <span><body><html>

#### Original toplevel document

Singular-value decomposition - Wikipedia
m , n ) , {\displaystyle T(\mathbf {V} _{i})=\sigma _{i}\mathbf {U} _{i},\qquad i=1,\ldots ,\min(m,n),} where σ i is the i-th diagonal entry of Σ, and T(V i ) = 0 for i > min(m,n). <span>The geometric content of the SVD theorem can thus be summarized as follows: for every linear map T : K n → K m one can find orthonormal bases of K n and K m such that T maps the i-th basis vector of K n to a non-negative multiple of the i-th basis vector of K m , and sends the left-over basis vectors to zero. With respect to these bases, the map T is therefore represented by a diagonal matrix with non-negative real diagonal entries. To get a more visual flavour of singular values and SVD factorization — at least when working on real vector spaces — consider the sphere S of radius one in R n . The linear map T map

#### Annotation 1731457584396

 #matrix In linear algebra, a square matrix A is called diagonalizable if it is similar to a diagonal matrix, i.e., if there exists an invertible matrix P such that P−1AP is a diagonal matrix.

Diagonalizable matrix - Wikipedia

#### Flashcard 1731460205836

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#matrix
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a square matrix A is called diagonalizable if it is similar to [...]

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In linear algebra, a square matrix A is called diagonalizable if it is similar to a diagonal matrix, i.e., if there exists an invertible matrix P such that P −1 AP is a diagonal matrix.

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Diagonalizable matrix - Wikipedia

#### Annotation 1731463089420

 #english Novelist Adam Langer skewers the publishing trade — and some of its recent trends — while digging toward something deeper.

Book review: 'The Thieves of Manhattan' by Adam Langer - latimes

#### Annotation 1731518926092

 #matrix-inversion A computationally simple and accurate way to compute the pseudoinverse is by using the singular value decomposition.[1][9][15] If is the singular value decomposition of A , then .

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A computationally simple and accurate way to compute the pseudoinverse is by using the singular value decomposition. [1] [9] [15] If is the singular value decomposition of A , then . For a rectangular diagonal matrix such as Σ {\displaystyle \Sigma } , we get the pseudoinverse by taking the reciprocal of each non-zero element on the diagonal, leaving the zeros in p

#### Original toplevel document

Moore–Penrose inverse - Wikipedia
A {\displaystyle A} and A ∗ {\displaystyle A^{*}} . Singular value decomposition (SVD)[edit source] <span>A computationally simple and accurate way to compute the pseudoinverse is by using the singular value decomposition. [1] [9] [15] If A = U Σ V ∗ {\displaystyle A=U\Sigma V^{*}} is the singular value decomposition of A, then A + = V Σ + U ∗ {\displaystyle A^{+}=V\Sigma ^{+}U^{*}} . For a rectangular diagonal matrix such as Σ {\displaystyle \Sigma } , we get the pseudoinverse by taking the reciprocal of each non-zero element on the diagonal, leaving the zeros in place, and then transposing the matrix. In numerical computation, only elements larger than some small tolerance are taken to be nonzero, and the others are replaced by zeros. For example, in the MATLAB, GNU Octave, or NumPy function pinv , the tolerance is taken to be t = ε⋅max(m,n)⋅max(Σ), where ε is the machine epsilon. The computational cost of this method is dominated by the cost of computing the SVD, which is several times higher than matrix–matrix multiplication, even if a state-of-the art implem

#### Flashcard 1731520498956

Tags
#matrix-inversion
Question
If is the singular value decomposition of A , then the pseudoinverse of A is [...]
.

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A computationally simple and accurate way to compute the pseudoinverse is by using the singular value decomposition. [1] [9] [15] If is the singular value decomposition of A , then .

#### Original toplevel document

Moore–Penrose inverse - Wikipedia
A {\displaystyle A} and A ∗ {\displaystyle A^{*}} . Singular value decomposition (SVD)[edit source] <span>A computationally simple and accurate way to compute the pseudoinverse is by using the singular value decomposition. [1] [9] [15] If A = U Σ V ∗ {\displaystyle A=U\Sigma V^{*}} is the singular value decomposition of A, then A + = V Σ + U ∗ {\displaystyle A^{+}=V\Sigma ^{+}U^{*}} . For a rectangular diagonal matrix such as Σ {\displaystyle \Sigma } , we get the pseudoinverse by taking the reciprocal of each non-zero element on the diagonal, leaving the zeros in place, and then transposing the matrix. In numerical computation, only elements larger than some small tolerance are taken to be nonzero, and the others are replaced by zeros. For example, in the MATLAB, GNU Octave, or NumPy function pinv , the tolerance is taken to be t = ε⋅max(m,n)⋅max(Σ), where ε is the machine epsilon. The computational cost of this method is dominated by the cost of computing the SVD, which is several times higher than matrix–matrix multiplication, even if a state-of-the art implem

#### Annotation 1731523644684

 pseudo datapoint based approximation methods for DGPs trade model complexity for a lower computational complexity of $$O(NLM^ 2 )$$ where N is the number of datapoints, L is the number of layers, and M is the number of pseudo datapoints.

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pseudo datapoint based approximation methods for DGPs trade model complexity for a lower computational complexity of $$O(NLM^ 2 )$$ where N is the number of datapoints, L is the number of layers, and M is the number of pseudo datapoints. This complexity scales quadratically in M whereas the dependence on the number of layers L is only linear. Therefore, it can be cheaper to increase the representation power of the model

#### Original toplevel document (pdf)

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#### Flashcard 1731525217548

Tags
#deep-gaussian-process
Question
pseudo datapoint based approximation methods for DGPs has a computational complexity of [...]
$$O(NLM^ 2 )$$

where N is the number of datapoints, L is the number of layers, and M is the number of pseudo datapoints.

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pseudo datapoint based approximation methods for DGPs trade model complexity for a lower computational complexity of $$O(NLM^ 2 )$$ where N is the number of datapoints, L is the number of layers, and M is the number of pseudo datapoints.

#### Original toplevel document (pdf)

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#### Flashcard 1731526790412

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#deep-gaussian-process
Question
DGPs can perform [...] or dimensionality compression or expansion
input warping

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DGPs can perform input warping or dimensionality compression or expansion, and automatically learn to construct a kernel that works well for the data at hand. As a result, learning in this model provides a flexible f

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#### Flashcard 1731528363276

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#deep-gaussian-process
Question
DGPs can perform input warping or [...]
dimensionality compression or expansion

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DGPs can perform input warping or dimensionality compression or expansion, and automatically learn to construct a kernel that works well for the data at hand. As a result, learning in this model provides a flexible form of Bayesian kernel design. </sp

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#### Flashcard 1731529936140

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#deep-gaussian-process
Question
DGPs can automatically learn to [...] that works well for the data at hand.
construct a kernel

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DGPs can perform input warping or dimensionality compression or expansion, and automatically learn to construct a kernel that works well for the data at hand. As a result, learning in this model provides a flexible form of Bayesian kernel design.

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#### Flashcard 1731531509004

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#deep-gaussian-process
Question
The new method uses an [...] procedure and a novel and efficient extension of the probabilistic backpropagation algorithm for learning.
approximate Expectation Propagation

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The new method uses an approximate Expectation Propagation procedure and a novel and efficient ex- tension of the probabilistic backpropagation algorithm for learning.

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#### Flashcard 1731533081868

Question
Deep Gaussian processes (DGPs) are [...] of Gaussian processes (GPs) and are formally equivalent to neural networks with multiple, infinitely wide hidden layers.
multi-layer hierarchical generalisations

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Deep Gaussian processes (DGPs) are multi-layer hierarchical generalisations of Gaussian pro- cesses (GPs) and are formally equivalent to neural networks with multiple, infinitely wide hidden layers.

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#### Flashcard 1731535703308

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#variational-inference
Question
Variational Bayesian methods are a family of techniques for approximating [...] arising in Bayesian inference and machine learning.
intractable integrals

In Bayesian inference this manifests as calculating marginal posteriors

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Variational Bayesian methods are a family of techniques for approximating intractable integrals arising in Bayesian inference and machine learning.

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Variational Bayesian methods - Wikipedia
f references, but its sources remain unclear because it has insufficient inline citations. Please help to improve this article by introducing more precise citations. (September 2010) (Learn how and when to remove this template message) <span>Variational Bayesian methods are a family of techniques for approximating intractable integrals arising in Bayesian inference and machine learning. They are typically used in complex statistical models consisting of observed variables (usually termed "data") as well as unknown parameters and latent variables, with various

#### Flashcard 1731550383372

Question
Artículo 114.- Ningún Senador o Representante, desde el día de su elección hasta el de su cese, podrá ser acusado criminalmente, ni aun por delitos comunes que no sean de los detallados en el artículo 93, sino ante su respectiva Cámara, la cual, por dos tercios de votos del total de sus componentes, resolverá si hay lugar a la formación de causa, y, en caso afirmativo, lo declarará suspendido en sus funciones y quedará a disposición del Tribunal competente.