Partial Identifiability
for NMF

Samyar Modabber

TU Chemnitz

2023/02/08

Content

Introduction

Exact NMF

Full Identifability

Partial Identifability

Part 1:

Introduction

LDR (Linear dimensionality reduction)

Find small number $r$ of basis vectors such that
each data point is well approximated by a linear combination of these basis vectors.

• Input: $\left\{ {{x}_{j}}\in {{\mathbb{R}}^{m}} \right\}_{j=1}^{n}$
• Output: $\left\{ {{w}_{k}}\in {{\mathbb{R}}^{m}} \right\}_{k=1}^{r}$
• Goal: Min $r$ such that ${{x}_{j}}\approx \sum\limits_{k=1}^{r}{{{w}_{k}}{{h}_{kj}}}$

Geometrical interpretation of LDR

LRMA (low-rank matrix approximation)

• Input: $X\in{{\mathbb{R}}^{m\times n}}$
• Output: $W\in {{\mathbb{R}}^{m\times r}}, H \in {{\mathbb{R}}^{r\times n}}$
• Goal: MIN $r$ such that $X\approx WH$

LDR is equivalent to LRMA

\[\begin{matrix} {} & LDR & \Leftrightarrow & LRMA \\ \forall j\in {{\mathbb{N}}_{n}} & {{x}_{j}} & {} & X\left( :,j \right) \\ \forall k\in {{\mathbb{N}}_{r}} & {{w}_{k}} & {} & W\left( :,k \right) \\ \forall j\in {{\mathbb{N}}_{n}} & {{h}_{j}} & {} & H\left( :,j \right) \\ \end{matrix}\]

NMF (Nonnegative matrix factorization)

• Input:
  nonnegetive matrix: $X\in \mathbb{R}_{+}^{m\times n}$ ,
  factorization rank: $r$,
  distance measure: $D:\ \mathbb{R}_{+}^{m\times n}\times \mathbb{R}_{+}^{m\times n}\to {{\mathbb{R}}_{+}}$
• Output: $W\in \mathbb{R}_{+}^{m\times r},\ H\in \mathbb{R}_{+}^{r\times n}$
• Goal: $$\underset{W,H}{\mathop{\min }}\,D\left( X,WH \right)$$

Basic Definitions

Let $A\in \mathbb{R}_{{}}^{m\times n}\ $ and ${{e}^{T}}=\left[ 1\ \cdots 1 \right]$, then we define:

• 1) Column space: $col\left( A \right):=\left\{ \left. Ay \right|y\in {{\mathbb{R}}^{n}} \right\}$
• 2) Affin hull: $aff\left( A \right):=\left\{ \left. Ay \right|y\in {{\mathbb{R}}^{n}},\ {{e}^{T}}y=1 \right\}$
• 3) Conical hull: $cone\left( A \right):=\left\{ \left. Ay \right|y\in {{\mathbb{R}}^{n}},\ y\ge 0 \right\}$
• 4) Convex hull: $conv\left( A \right):=\left\{ \left. Ay \right|y\in {{\mathbb{R}}^{n}},\ {{e}^{T}}y=1,\ y\ge 0 \right\}$
• 5) Support: $\operatorname{supp}\left( A \right)=\left\{ \left. \left( i,j \right) \right|A\left( i,j \right)\ne 0 \right\}$,
  for vectors $\operatorname{supp}\left( a \right)=\left\{ \left. j \right|{{a}_{j}}\ne 0 \right\}$
• 6) A is stochastic if ${{e}^{T}}A={{e}^{T}}$
• 7) Normalized of A: $\theta \left( A \right):=\left[ \begin{matrix}{}^{{{a}_{1}}}/{}_{{{\left\| {{a}_{1}} \right\|}_{1}}} & \cdots & {}^{{{a}_{n}}}/{}_{{{\left\| {{a}_{n}} \right\|}_{1}}} \\ \end{matrix} \right]$
• 8) Nonnegative Orthant: $\mathbb{R}_{+}^{n}:=\left\{ \left. y \right|y\in {{\mathbb{R}}^{n}},\ y\ge 0 \right\}$
• 9) Probability Simplex: $\Delta ={{\Delta }^{n}}:=\left\{ \left. y \right|y\in {{\mathbb{R}}^{n}},\ {{e}^{T}}y=1,\ y\ge 0 \right\}=conv\left( I \right)$

Part 2:

Exact NMF

Exact NMF

Decomposition $R=C{{S}^{T}}$ of $R\in \mathbb{R}_{+}^{m\times n}$ is called
an Exact NMF of size $r$ and we write $R\ \overset{Exact}{\mathop{=}}\,C{{S}^{T}}$ if:

$C\in \mathbb{R}_{+}^{m\times r},\ S\in \mathbb{R}_{+}^{n\times r}$

Fact

For an Exact NMF $R\overset{Exact}{\mathop{=}}\,C{{S}^{T}}$, w.l.o.g.
we can assume $R,\ C\ and \ {{S}^{T}}$ are stochastic .

Proof: First remove zero columns then define

$$\begin{align*} {{R}_{n}}\left( :,j \right) &:=\frac{R\left( :,j \right)}{{{e}^{T}}R\left( :,j \right)}=\frac{\sum\limits_{k=1}^{r}{C\left( :,k \right)S\left( j,k \right)}}{{{e}^{T}}R\left( :,j \right)} \\ & =\sum\limits_{k=1}^{r}{\left( \frac{C\left( :,k \right)}{{{e}^{T}}C\left( :,j \right)} \right)\left( \frac{{{e}^{T}}C\left( :,j \right)}{{{e}^{T}}R\left( :,j \right)}S\left( j,k \right) \right)} \\ & =\sum\limits_{k=1}^{r}{{{C}_{n}}\left( :,k \right){{S}_{n}}\left( j,k \right)} \\ \end{align*}$$

Hence ${{R}_{n}}$and ${{C}_{n}}$are stochastic and for $S_{n}^{T}$: ${{e}^{T}}S_{n}^{T}=\left( {{e}^{T}}{{C}_{n}} \right)S_{n}^{T}={{e}^{T}}{{R}_{n}}={{e}^{T}}$

NPP (Nested Polytope Problem)

Input: ${{P}_{inn}},\ {{P}_{out}}$ such that:

1) full-dimensional inner polytope: ${{P}_{inn}}:=\ conv\left( \left[ {{v}_{1}},{{v}_{2}},\cdots ,{{v}_{n}} \right] \right)\subseteq {{\mathbb{R}}^{d}}$

2) full-dimensional outer polytope: ${{P}_{out}}:=\ \left\{ \left. x\in {{\mathbb{R}}^{d}} \right|Fx+g\ge 0 \right\}$
where $F\in \mathbb{R}_{{}}^{m\times r},\ g\in {{\mathbb{R}}^{m}}$

Output: ${{P}_{bet}}$ with $p$ vertices such that $p\ge d+1$

Goal: ${{P}_{inn}}\subseteq {{P}_{bet}}\subseteq {{P}_{out}}$

Theorem 1

An Exact NMF problem with $r\text{ }=\text{ }rank\left( R \right)$

$$\Leftrightarrow $$ $$\Leftrightarrow $$ $$\Leftrightarrow $$

An NPP with $d\text{ }=\text{ }rank\left( R \right)\text{ }-\text{ }1$
and $p\text{ }=\text{ }d\text{ }+\text{ }1$.

Example 1

${{P}_{inn}}:=\ conv\left( \left[ {{v}_{1}}=>\left[ \begin{matrix} 0.5 \\ 0 \\ \end{matrix} \right],{{v}_{2}}=\left[ \begin{matrix} 0 \\ 0.5 \\ \end{matrix} \right],{{v}_{3}}=\left[ \begin{matrix} 0.25 \\ 0.75 \\ \end{matrix} \right],{{v}_{4}}=\left[ \begin{matrix} 0.75 \\ 0.25 \\ \end{matrix} \right] \right] \right)\subseteq {{\mathbb{R}}^{2}}$

${{P}_{out}}:=\ \left\{ \left. x\in {{\mathbb{R}}^{2}} \right|Fx+g={{\left[ \begin{matrix} 0 & 0 & 1 & -1 \\ 1 & -1 & 0 & 0 \\ \end{matrix} \right]}^{T}}x+{{\left[ \begin{matrix} 0 & 1 & 0 & 1 \\ \end{matrix} \right]}^{T}}\ge 0 \right\}$

$R\left( :,j \right)=F{{v}_{j}}+g=\frac{1}{4}\left[ \begin{matrix} 0 & 2 & 3 & 1 \\ 4 & 2 & 1 & 3 \\ 2 & 0 & 1 & 3 \\ 2 & 4 & 3 & 1 \\ \end{matrix} \right]$

${{P}_{bet}}:=\ conv\left( \left[ {{s}_{1}}=\left[ \begin{matrix} 0 \\ 0 \\ \end{matrix} \right],{{s}_{2}}=\left[ \begin{matrix} 0 \\ 1 \\ \end{matrix} \right],{{s}_{3}}=\left[ \begin{matrix} 1 \\ 0 \\ \end{matrix} \right] \right] \right)$

$C\left( :,j \right)=F{{s}_{j}}+g=\left[ \begin{matrix} 0 & 0 & 1 \\ 1 & 1 & 0 \\ 0 & 1 & 0 \\ 1 & 0 & 1 \\ \end{matrix} \right]$

Illustration of the NPP (Example 1)

Part 3:

Full
identifiability

Full identifiability

Exact NMF $R\ \overset{Exact}{\mathop{=}}\,{{C}_{*}}S_{*}^{T}$ of $R\in \mathbb{R}_{+}^{m\times n}$ is called
a (full) identifiability of rank $r$
and we write $R\ \overset{Full}{\mathop{=}}\,C{{S}^{T}}$ if:

$$\forall \ R \: \overset{Exact}{\mathop{=}}\: C{{S}^{T}}\ \exists \: \pi :{{\mathbb{N}}_{r}}\to {{\mathbb{N}}_{r}}\ \exists \: 0 < \alpha \in {{\mathbb{R}}^{r}}$$ $$C\left( :,k \right)={{\alpha }_{k}}{{C}_{*}}\left( :,{{\pi }_{k}} \right)\ ,\ \ S\left( :,k \right)^{T}=\frac{1}{{{\alpha }_{k}}}{{S}_{*}}\left( :,{{\pi }_{k}} \right)^{T}$$

Remark

Full identifiability with Matrix notation

$\forall \ R\overset{Exact}{\mathop{=}}\,C{{S}^{T}}$, there exists
a permutation matrix $\Pi \in {{\left\{ 0,1 \right\}}^{r\times r}}$ and
a nonsingular diagonal scaling matrix $D$
such that \[C={{C}_{*}}\Pi D\ \wedge \ {{S}^{T}}={{D}^{-1}}{{\Pi }^{T}}S_{*}^{T}\]

Theorem 2

$$R\ \overset{Full}{\mathop{=}}\,{{C}_{*}}S_{*}^{T}\ \Rightarrow \ \ $$ $ \forall j,{j}'\ \ j\ne {j}'\ \ \ \operatorname{supp}\left( {{C}_{*}}\left( :,j \right) \right)\not\subset \operatorname{supp}\left( {{C}_{*}}\left( :,{j}' \right) \right)$

Separability

$C\in \mathbb{R}_{{}}^{m\times r}$ with $m\text{ }\ge \text{ }r$ is called separable if

There exists an index set $K$ of size $r$ such that $C\left( K,: \right)\in \mathbb{R}_{{}}^{r\times r}$ is a nonsingular diagonal matrix.

Theorem 3

The Exact NMF $R\overset{Exact}{\mathop{=}}\,{{C}_{*}}S_{*}^{T}$
is a Full Identifiability
if ${{C}_{*}}$ and $S_{*}^{T}$ are separable.

SSC (sufficiently scattered condition)

$C\in \mathbb{R}_{{}}^{m\times r}$ with $m \ge r$ satisfies $SSC$ if:

1) $L:=\left\{ \left. x\in \mathbb{R}_{+}^{r} \right|{{e}^{T}}x\ge \sqrt{r-1}{{\left\| x \right\|}_{2}} \right\}\subseteq cone\left( {{C}^{T}} \right)$
$L$ is a Lorentz cone, also known as a second-order cone or an ice cream cone.

2) Does not exist any orthogonal matrix $Q$ ($Q{{Q}^{T}}=I\ $) such that $$cone\left( {{C}^{T}} \right)\subseteq cone\left( Q \right),$$ except for permutation matrices.

Theorem 4

The Exact NMF $R\overset{Exact}{\mathop{=}}\,{{C}_{*}}S_{*}^{T}$
is a Full Identifiability
if ${{C}_{*}}$ and $S_{*}^{T}$ satisfy the SSC.

Separability vs. SSC

Part 4:

Partial
Identifiability

Partial identifiability

Let $R\ \overset{Exact}{\mathop{=}}\,{{C}_{*}}S_{*}^{T}$. ${{C}_{*}}\left( :,k \right)$ is called identifiable if

$\forall \ R\overset{Exact}{\mathop{=}}\,C{{S}^{T}}\ \exists j\ \ \exists \alpha >0\ \ \ \ {{C}_{*}}\left( :,k \right)=\alpha C\left( :,j \right)$

Theorem 5

If $R\ \overset{\ Exact}{\mathop{=}}\,{{C}_{*}}S_{*}^{T}$ and ${{C}_{*}}\left( :,k \right)$is identifiable,
then

$$\ \forall j\ \ j\ne k\ \ \operatorname{supp}\left( {{C}_{*}}\left( :,j \right) \right)\not\subset \operatorname{supp}\left( {{C}_{*}}\left( :,k \right) \right)$$ $ \left( resp.\ {{S}_{*}} \right)$

Theorem 6

Let $R\ \overset{Exact}{\mathop{=}}\,{{C}_{*}}S_{*}^{T}$ of size $r$.
Then ${{C}_{*}}\left( :,k \right)$ is identifiable if:

1) FRZRW (Full-rank zero-region window):
$$rank\left( {{C}_{*}}\left( I,: \right) \right)=r-1$$ where $I=\left\{ \left. i \right|{{C}_{*}}\left( i,k \right)=0 \right\}=\text{supp}{{\left( C\left( :,k \right) \right)}^{c}}$

2) SW (Selective window):
$$\exists j\ \exists \alpha >0\ \ \ \ {{S}_{*}}\left( j,: \right)=\alpha e_{\left( k \right)}^{T}$$

Example 2

$R\ \overset{Exact}{\mathop{=}}\,{{C}_{*}}S_{*}^{T}=\left[ \begin{matrix} 2 & 2 & 2 \\ 1 & 3 & 1 \\ 1 & 1 & 3 \\ 0 & 2 & 2 \\ 0 & 1 & 2 \\ \end{matrix} \right]\left[ \begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{matrix} \right]$

FRZRW:
$I=\left\{ 4,5 \right\}\ \Rightarrow C\left( I,: \right)=\left[ \begin{matrix} 0 & 2 & 2 \\ 0 & 1 & 2 \\ \end{matrix} \right]\ \\ \Rightarrow rank\left( C\left( I,: \right) \right)=2=3-1=r-1$

SW:
$\exists j=1\ \exists \alpha =1>0\ \ \ \ {{S}_{*}}\left( 1,: \right)=e_{\left( 1 \right)}^{T}$

Minimal face

Let $C\in \mathbb{R}_{+}^{m\times r}$ be a stochastic matrix and
$y\in \Gamma =col\left( C \right)\bigcap \Delta $.
We define minimal face of $C$ containing $y$ by

$$ \begin{equation*} \begin{aligned} {{F}_{C}}\left( y \right) & =\left\{ \left. x\in \Gamma \right|\text{supp}\left( x \right)\subseteq \text{supp}\left( y \right) \right\} \\ & =\left\{ \left. Cz \right|\text{z}\in {{\mathbb{R}}^{r}}\text{,}\ \ {{\text{e}}^{T}}\text{z=1}\text{,}\ {{y}_{i}}=0\Rightarrow {{\left( Cz \right)}_{i}}=0 \right\} \end{aligned} \end{equation*} $$

All the points in ${{F}_{C}}\left( y \right)$ have to belong to the same facets of $C$ as $y$.

Note

1. A face of a polytope is obtained by intersecting a subset of its facets.
   Hence ${{F}_{C}}\left( y \right)$ is the face of $C$ of minimal dimension containing $y$.
2. A vertex of a poltyope is a 0-dimensional face, and hence
   $y$ is a vertex of $C$ $\Leftrightarrow {{F}_{C}}\left( y \right)=\left\{ y \right\}$.

Lemma 1

Let $C\in \mathbb{R}_{+}^{m\times r}$ be a nonsingular stochastic matrix.

${{C}_{*}}\left( :,k \right)$ has FRZRW condition
$\Leftrightarrow $
$${{F}_{C}}\left( C\left( :,k \right) \right)=\left\{ C\left( :,k \right) \right\}$$

Theorem 7

Let $R\ \overset{Exact}{\mathop{=}}\,{{C}_{*}}S_{*}^{T}$ with $r=rank\left( R \right)$,
${{C}_{*}}\in \mathbb{R}_{+}^{m\times r}$ and $S_{*}^{T}\in \mathbb{R}_{+}^{n\times r}$ are stochastic.

1) it satisfies the selective window condition

2) there exists index set $J$such that $$rank\left( R\left( :,J \right) \right)=r-1\ \\ \\ \ \forall j\in J\ {{F}_{{{C}_{*}}}}\left( {{C}_{*}}\left( :,k \right) \right)\bigcap {{F}_{{{C}_{*}}}}\left( R\left( :,j \right) \right)=\varnothing $$

Example 3

${{P}_{inn}}:=\ conv\left( \left[ {{v}_{1}}=\left[ \begin{matrix} 0.5 \\ 0 \\ \end{matrix} \right],{{v}_{2}}=\left[ \begin{matrix} 0.2 \\ 1 \\ \end{matrix} \right],{{v}_{3}}=\left[ \begin{matrix} 0.8 \\ 1 \\ \end{matrix} \right] \right] \right)\subseteq {{\mathbb{R}}^{2}}$

${{P}_{out}}:=\ \left\{ \left. x\in {{\mathbb{R}}^{2}} \right|Fx+g={{\left[ \begin{matrix} 0 & 0 & 1 & -1 \\ 1 & -1 & 0 & 0 \\ \end{matrix} \right]}^{T}}x+{{\left[ \begin{matrix} 0 & 1 & 0 & 1 \\ \end{matrix} \right]}^{T}}\ge 0 \right\}$

$R=\left[ \begin{matrix} 0 & 1 & 1 \\ 1 & 0 & 0 \\ 0.5 & 0.2 & 0.8 \\ 0.5 & 0.8 & 0.2 \\ \end{matrix} \right]=\left[ \begin{matrix} 0 & 1 & 1 \\ 1 & 0 & 0 \\ 0.5 & 0 & 1 \\ 0.5 & 1 & 0 \\ \end{matrix} \right]\left[ \begin{matrix} 1 & 0 & 0 \\ 0 & 0.8 & 0.2 \\ 0 & 0.2 & 0.8 \\ \end{matrix} \right]={{C}_{*}}S_{*}^{T}$

$C\left( :,1 \right)$ satisfies the selective window and minimal faces of $R\left( :,\left[ 2,3 \right] \right)$do not intersect with minimal faces of $C\left( :,1 \right)$

Illustration of the NPP (Example 3)

Corollary 1

Let $R\ \overset{Exact}{\mathop{=}}\,{{C}_{*}}S_{*}^{T}$ with $r=rank\left( R \right)$,
${{C}_{*}}\in \mathbb{R}_{+}^{m\times r}$ and $S_{*}^{T}\in \mathbb{R}_{+}^{n\times r}$ are stochastic.
Then $R\ \overset{\ Full}{\mathop{=}}\,{{C}_{*}}S_{*}^{T}$ if

1) $\forall k\ \ {{C}_{*}}\left( :,k \right)$satisfies the selective window.

2) $\forall k,j\ \ \ k\ne j\Rightarrow \ {{F}_{{{C}_{*}}}}\left( {{C}_{*}}\left( :,k \right) \right)\bigcap {{F}_{{{C}_{*}}}}\left( {{C}_{*}}\left( :,j \right) \right)=\varnothing $

Example 4

${{P}_{inn}}:=\ conv\left( \left[ {{v}_{1}}=\left[ \begin{matrix} 0 \\ 0 \\ 0.75 \\ \end{matrix} \right],{{v}_{2}}=\left[ \begin{matrix} 0 \\ 1 \\ 0.25 \\ \end{matrix} \right],{{v}_{3}}=\left[ \begin{matrix} 1 \\ 1 \\ 0.75 \\ \end{matrix} \right] \right] \right)\subseteq {{\mathbb{R}}^{3}}$

${{P}_{out}}:=\ {{\left[ 0,1 \right]}^{3}}$

satisfies the conditions of Theorem 7 for all $k$

since the vertices of ${{P}_{inn}}$ are on (minimal) faces (namely, edges) that do not intersect.

therefore R has a unique exact NMF

Illustration of the NPP (Example 4)

Theorem 8

Let $R\ \overset{Exact}{\mathop{=}} {{C}_{*}}S_{*}^{T}$ with $r=rank\left( R \right)$, ${{C}_{*}}\in \mathbb{R}_{+}^{m\times r}$ and $S_{*}^{T}\in \mathbb{R}_{+}^{n\times r}$
and w.l.o.g $\forall j=1,\ 2,\ ...,\ p;\ {{C}_{*}}\left( :,j \right)$ are identifiable.

Let $J$ be the index set of columns of $R$
that do not contain the support of the first $p$ columns of ${{C}_{*}}$. and

1. $rank\left( {{S}_{*}}\left( J,p+1:r \right) \right)=r-p$
2. ${{C}_{*}}\left( :,p+1 \right)$ identifiable in
   $R\left( :J \right)\overset{Exact}{\mathop{=}}\,{{C}_{*}}\left( :,p+1:r \right){{S}_{*}}{{\left( J,p+1:r \right)}^{T}}$of size $r-p$

then ${{C}_{*}}\left( :,p+1 \right)$ is identifiable in the Exact NMF of $R$ of size $r$.

Lemma 2

Let $R\ \overset{\ Exact}{\mathop{=}}\,{{C}_{*}}S_{*}^{T}$ and $R\ \overset{Exact}{\mathop{=}}\,C{{S}^{T}}$ with size $r=rank\left( R \right)$ and w.l.o.g $C$ and ${{C}_{*}}$ are stochastic. If ${{C}_{*}}\left( :,k \right)$ satisfied following condition:

1) The selective window assumption, i.e., $\exists \alpha >0\ \ \exists j\ \ \ st\ {{C}_{*}}\left( :,k \right)=\alpha R\left( :,j \right)$
2) Is not identifiable in $C$, i.e. $\forall j\ \ \ C\left( :,j \right)\ne {{C}_{*}}\left( :,k \right)$

Then there exists an index set $J$ with $\left| J \right|\ge 2$ such that
$$\forall j\in J\ \ \ \ C\left( :,j \right)\in {{F}_{{{C}_{*}}}}\left( {{C}_{*}}\left( :,k \right) \right)$$

Theorem 9

Let $R\ ={{C}_{*}}S_{*}^{T}$ with size $r=3=rank\left( R \right)$, ${{C}_{*}}\in \mathbb{R}_{+}^{m\times 3}$ and $S_{*}^{T}\in \mathbb{R}_{+}^{n\times 3}$ and w.l.o.g $R$, ${{C}_{*}}$ and $S_{*}^{T}$ are stochastic.

W.l.o.g let ${{C}_{*}}\left( :,1 \right)$ and ${{C}_{*}}\left( :,2 \right)$satisfied selective window and their support do not contain each other.

Then these two columns are identifiable if $\exists R\left( :,j \right)$ such that:

IF ${{F}_{{{C}_{*}}}}\left( {{C}_{*}}\left( :,1 \right) \right)\bigcap {{F}_{{{C}_{*}}}}\left( {{C}_{*}}\left( :,2 \right) \right)=\varnothing $,
\[R\left( :,j \right)\notin \ conv\left( \left[ {{C}_{*}}\left( :,1 \right),{{F}_{{{C}_{*}}}}\left( {{C}_{*}}\left( :,2 \right) \right) \right] \right)\bigcup conv\left( \left[ {{C}_{*}}\left( :,2 \right),{{F}_{{{C}_{*}}}}\left( {{C}_{*}}\left( :,1 \right) \right) \right] \right)\]

Else
\[R\left( :,j \right)\notin \ conv\left( \left[ {{F}_{{{C}_{*}}}}\left( {{C}_{*}}\left( :,1 \right) \right),{{F}_{{{C}_{*}}}}\left( {{C}_{*}}\left( :,2 \right) \right) \right] \right)\]

Algotithm

Refrences

• N. Gillis and R. Rajkó, "Partial Identifiability for Nonnegative Matrix Factorization", SIAM J. on Matrix Analysis and Applications 44 (1), pp. 27-52, 2023.
• N. Gillis, "Nonnegative Matrix Factorization", SIAM, Philadelphia, 2020.

Thank you

Please find the presentation on my GitHub: https://github.com/samyarmodabber/TUC_Seminar_NMF