Q stringlengths 4 3.96k | A stringlengths 1 3k | Result stringclasses 4
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Lemma 8.2. Let \( \\mathbf{A} \) and \( \\mathbf{U} \) be defined as in Lemma 8.1 and \( \\mathbf{Q} \) be an \( \\left( {n \\times n}\\right) \) - dimensional matrix which is symmetric and positive definite. Then; we have\n\n\[ \n\\mathbf{Q}\\left\\lbrack {\\mathbf{I} - {\\mathbf{{QA}}}^{T}{\\left( {\\mathbf{{AQ}}}^{2... | Proof. Since \( \\mathbf{Q} \) is positive definite, \( {\\mathbf{Q}}^{-1} \) exists. If we define \( \\widehat{\\mathbf{A}} = \\mathbf{{AQ}} \) and \( \\widehat{\\mathbf{U}} = \) \( {\\mathbf{Q}}^{-1}\\mathbf{U} \), then \( \\widehat{\\mathbf{A}} \) is an \( m \\times n \) matrix with full row rank and \( \\widehat{\\... | Yes |
\[ \text{Minimize}2{x}_{1}^{2} + 3{x}_{2}^{2} + 5{x}_{3}^{2} + {x}_{1} + 2{x}_{2} - 3{x}_{3} \] \[ \text{subject to}\;{x}_{1} + {x}_{2} = 5 \] \[ {x}_{2} + {x}_{3} = {10} \] \[ {x}_{1},{x}_{2},{x}_{3} \geq 0 \] | It is clear that \[ \mathbf{A} = \left\lbrack \begin{array}{lll} 1 & 1 & 0 \\ 0 & 1 & 1 \end{array}\right\rbrack ,\;\mathbf{b} = \left\lbrack \begin{array}{l} 5 \\ {10} \end{array}\right\rbrack ,\;\mathbf{c} = \left\lbrack \begin{array}{r} 1 \\ 2 \\ - 3 \end{array}\right\rbrack ,\;\text{ and }\;\mathbf{Q} = \left\lbrac... | Yes |
Theorem 10.2. Let \( \mathbf{Q} \) be an \( \left( {m \times n}\right) \) -dimensional orthonormal matrix with full row-rank \( m < n \) ; then \( \parallel \mathbf{Q}\parallel \leq 1 \) and \( \begin{Vmatrix}{\mathbf{Q}}^{T}\end{Vmatrix} = 1 \) . | Proof. Since \( \mathbf{Q} \) is orthonormal with row-rank \( m < n \), by the Gram-Schmidt orthogonalization process, there exists an \( \left\lbrack {\left( {n - m}\right) \times n}\right\rbrack \) -dimensional matrix \( \mathbf{R} \) such that\n\n\[ \overline{\mathbf{Q}} = \left\lbrack \begin{array}{l} \mathbf{Q} \\... | Yes |
Theorem 10.3 (fundamental theorem for LQ factorization). Let \( \mathrm{A} \) be an \( \left( {m \times n}\right) \) -dimensional matrix with full row-rank \( m < n \) ; then there exists an \( \left( {m \times m}\right) \) - dimensional lower triangular matrix \( \mathbf{L} \) and an \( \left( {m \times n}\right) \) -... | Proof. Since A has full row-rank with \( m < n \), by Theorem 10.1, we know that \( {\mathbf{{AA}}}^{T} = {\mathbf{{LL}}}^{T} \) for a lower triangular matrix \( \mathbf{L} \) with positive diagonal elements. Hence \( {\mathbf{L}}^{-1} \) exists and \( \mathbf{Q} = {\mathbf{L}}^{-1}\mathbf{A} \) is well defined. Moreov... | Yes |
Theorem 10.4. The sequence \( {\mathbf{w}}^{\left( j\right) } : j = 0,1,2,\ldots \) generated by Algorithm LQ- 2 is a Cauchy sequence and thus converges. Moreover, if we let \( \mathbf{w} \) be its limit point, then \( {\mathbf{w}}^{k} = \left( {1/\alpha }\right) \mathbf{w} \) solves the system (10.32). | Proof. Let \( p \geq 1 \), then\n\n\[ \begin{aligned} \left| \left| {{\mathbf{w}}^{\left( j + p\right) } - {\mathbf{w}}^{\left( j\right) }}\right| \right| & \leq \left| \left| {{\mathbf{w}}^{\left( j + p\right) } - {\mathbf{w}}^{\left( j + p - 1\right) }}\right| \right| + \cdots + \left| \left| {{\mathbf{w}}^{\left( j ... | Yes |
Proposition 2.1 (Kantorovich for matching). If \( m = n \) and \( \mathbf{a} = \mathbf{b} = {\mathbb{1}}_{n}/n \), then there exists an optimal solution for Problem (2.11) \( {\mathbf{P}}_{{\sigma }^{ \star }} \), which is a permutation matrix associated to an optimal permutation \( {\sigma }^{ \star } \in \operatornam... | Proof. Birkhoff’s theorem [1946] states that the set of extremal points of \( \mathbf{U}\left( {{\mathbb{1}}_{n}/n,{\mathbb{1}}_{n}/n}\right) \) is equal to the set of permutation matrices. A fundamental theorem of linear programming [Bertsimas and Tsitsiklis, 1997, Theorem 2.7] states that the minimum of a linear obje... | Yes |
Proposition 2.3. We assume \( \mathcal{X} = \mathcal{Y} \) and that for some \( p \geq 1, c\left( {x, y}\right) = d{\left( x, y\right) }^{p} \) , where \( d \) is a distance on \( \mathcal{X} \), i.e.\n\n(i) \( d\left( {x, y}\right) = d\left( {y, x}\right) \geq 0 \) ;\n\n(ii) \( d\left( {x, y}\right) = 0 \) if and only... | Proof. The proof follows the same approach as that for Proposition 2.2 and relies on the existence of a coupling between \( \left( {\alpha ,\gamma }\right) \) obtained by \ | No |
Proposition 3.1. The following identities, in which the inequality sign between vectors should be understood elementwise, hold:\n\n(i) \( \mathbf{f} \leq {\mathbf{f}}^{\prime } \Rightarrow {\mathbf{f}}^{\mathbf{C}} \geq {\mathbf{f}}^{\prime \mathbf{C}} \) ,\n\n(ii) \( {\mathbf{f}}^{\mathrm{{CC}}} \geq \mathbf{f},{\math... | Proof. The first inequality follows from the definition of \( \mathbf{C} \)-transforms. Expanding the definition of \( {\mathbf{f}}^{\mathbf{{CC}}} \) we have\n\n\[ \n{\left( {\mathbf{f}}^{\mathbf{C}}\overline{\mathbf{C}}\right) }_{i} = \mathop{\min }\limits_{{j \in \llbracket m\rrbracket }}{\mathbf{C}}_{ij} - {\mathbf... | Yes |
Proposition 3.2. Let \( {\mathbf{P}}^{ \star } \) and \( {\mathbf{f}}^{ \star },{\mathbf{g}}^{ \star } \) be optimal solutions for the primal (2.24) and dual (2.11) problems, respectively. Then, for any pair \( \left( {i, j}\right) \in \llbracket n\rrbracket \times \llbracket m\rrbracket ,{\mathbf{P}}_{i, j}^{ \star }\... | Proof. We have by strong duality that \( \left\langle {{\mathbf{P}}^{ \star },\mathbf{C}}\right\rangle = \left\langle {{\mathbf{f}}^{ \star },\mathbf{a}}\right\rangle + \left\langle {{\mathbf{g}}^{ \star },\mathbf{b}}\right\rangle \) . Recall that \( {\mathbf{P}}^{ \star }{\mathbb{1}}_{m} = \) a and \( {\mathbf{P}}^{\s... | Yes |
Proposition 3.3. If \( \mathbf{P} \) and \( \left( {\mathbf{f},\mathbf{g}}\right) \) are complementary and feasible solutions for the primal (2.24) and dual (2.11) problems, respectively, then \( \mathbf{P} \) and \( \left( {\mathbf{f},\mathbf{g}}\right) \) are both primal and dual optimal. | Proof. By weak duality, we have that\n\n\[ \n{L}_{\mathbf{C}}\left( {\mathbf{a},\mathbf{b}}\right) \leq \langle \mathbf{P},\mathbf{C}\rangle = \langle \mathbf{P},\mathbf{f} \oplus \mathbf{g}\rangle = \langle \mathbf{a},\mathbf{f}\rangle + \langle \mathbf{b},\mathbf{g}\rangle \leq {L}_{\mathbf{C}}\left( {\mathbf{a},\mat... | Yes |
Proposition 3.4 (Extremal solutions). Let \( \mathbf{P} \) be an extremal point of the polytope \( \mathbf{U}\left( {\mathbf{a},\mathbf{b}}\right) \). Let \( S\left( \mathbf{P}\right) \subset \mathcal{E} \) be the subset of edges \( \left\{ {\left( {i,{j}^{\prime }}\right), i \in \llbracket n\rrbracket, j \in \llbracke... | Proof. We proceed by contradiction. Suppose that \( \mathbf{P} \) is an extremal point of the polytope \( \mathbf{U}\left( {\mathbf{a},\mathbf{b}}\right) \) and that its corresponding set \( S\left( \mathbf{P}\right) \) of edges, denoted \( F \) for short, is such that the graph \( G = \left( {V \cup {V}^{\prime }, F}\... | Yes |
Proposition 3.5. \( \left( {\widetilde{\mathbf{f}},\widetilde{\mathbf{g}}}\right) \overset{\text{ def. }}{ = }\left( {\mathbf{f},\mathbf{g}}\right) + \varepsilon \left( {{\mathbb{1}}_{S}, - {\mathbb{1}}_{{S}^{\prime }}}\right) \) is dual feasible for a small enough \( \varepsilon > 0 \) if for all \( i \in S \), the fa... | Proof. For any \( i \in S \), consider the set \( {\mathcal{I}}_{i} \) of all \( {j}^{\prime } \in \llbracket m{\rrbracket }^{\prime } \) such that \( \left( {i,{j}^{\prime }}\right) \) is inactive, namely such that \( {\mathbf{f}}_{i} + {\mathbf{g}}_{j} < {\mathbf{C}}_{ij} \) . Define \( {\varepsilon }_{i}\overset{\te... | Yes |
Proposition 3.7. The auction algorithm maintains \( \varepsilon \) -complementary slackness at each iteration. | Proof. Let \( \mathbf{g},\xi, S \) be the three variables at the beginning of a given iteration. We therefore assume that for any \( {i}^{\prime } \in S \) the relationship\n\n\[{\mathbf{C}}_{i,{\xi }_{{i}^{\prime }}} - {\mathbf{g}}_{{\xi }_{{i}^{\prime }}} \leq \varepsilon + \mathop{\min }\limits_{j}{\mathbf{C}}_{{i}^... | Yes |
Proposition 3.8. The number of steps of the auction algorithm is at most \( N = \) \( n\parallel \mathbf{C}{\parallel }_{\infty }/\varepsilon \) . | Proof. Suppose that the algorithm has not stopped after \( T > N \) steps. Then there exists an index \( j \) which is not in the image of \( \xi \), namely whose price coordinate \( {\mathbf{g}}_{j} \) has never been updated and is still \( {\mathbf{g}}_{j} = 0 \) . In that case, there cannot exist an index \( {j}^{\p... | Yes |
Proposition 3.9. The auction algorithm finds an assignment whose cost is \( {n\varepsilon } \) suboptimal. | Proof. Let \( \sigma ,{\mathbf{g}}^{ \star } \) be the primal and dual optimal solutions of the assignment problem of matrix \( \mathbf{C} \), with optimum\n\n\[ \n{t}^{ \star } = \sum {\mathbf{C}}_{i,{\sigma }_{i}} = \mathop{\sum }\limits_{i}\mathop{\min }\limits_{j}{\mathbf{C}}_{i, j} - {\mathbf{g}}_{j}^{ \star } + \... | Yes |
Proposition 4.1 (Convergence with \( \varepsilon \) ). The unique solution \( {\mathbf{P}}_{\varepsilon } \) of (4.2) converges to the optimal solution with maximal entropy within the set of all optimal solutions of the Kantorovich problem, namely\n\n\[ \n{\mathbf{P}}_{\varepsilon }\overset{\varepsilon \rightarrow 0}{ ... | Proof. We consider a sequence \( {\left( {\varepsilon }_{\ell }\right) }_{\ell } \) such that \( {\varepsilon }_{\ell } \rightarrow 0 \) and \( {\varepsilon }_{\ell } > 0 \) . We denote \( {\mathbf{P}}_{\ell } \) the solution of (4.2) for \( \varepsilon = {\varepsilon }_{\ell } \) . Since \( \mathbf{U}\left( {\mathbf{a... | Yes |
Proposition 4.2. For any \( \pi \in \mathcal{U}\left( {\alpha ,\beta }\right) \), and for any \( \left( {{\alpha }^{\prime },{\beta }^{\prime }}\right) \) having the same 0 measure sets as \( \left( {\alpha ,\beta }\right) \) (so that they have both densities with respect to one another) one has\n\n\[ \mathrm{{KL}}\lef... | This proposition shows that choosing \( \mathrm{{KL}}\left( {\cdot \mid {\alpha }^{\prime } \otimes {\beta }^{\prime }}\right) \) in place of \( \mathrm{{KL}}\left( {\cdot \mid \alpha \otimes \beta }\right) \) in (4.9) results in the same solution. | No |
Proposition 4.3. The solution to (4.2) is unique and has the form\n\n\[ \forall \left( {i, j}\right) \in \llbracket n\rrbracket \times \llbracket m\rrbracket ,\;{\mathbf{P}}_{i, j} = {\mathbf{u}}_{i}{\mathbf{K}}_{i, j}{\mathbf{v}}_{j} \]\n\nfor two (unknown) scaling variable \( \left( {\mathbf{u},\mathbf{v}}\right) \in... | Proof. Introducing two dual variables \( \mathbf{f} \in {\mathbb{R}}^{n},\mathbf{g} \in {\mathbb{R}}^{m} \) for each marginal constraint, the Lagrangian of (4.2) reads\n\n\[ \mathcal{E}\left( {\mathbf{P},\mathbf{f},\mathbf{g}}\right) = \langle \mathbf{P},\mathbf{C}\rangle - \varepsilon \mathbf{H}\left( \mathbf{P}\right... | Yes |
One has \( \left( {{\mathbf{u}}^{\left( \ell \right) },{\mathbf{v}}^{\left( \ell \right) }}\right) \rightarrow \left( {{\mathbf{u}}^{ \star },{\mathbf{v}}^{ \star }}\right) \) and\n\n\[ \n{d}_{\mathcal{H}}\left( {{\mathbf{u}}^{\left( \ell \right) },{\mathbf{u}}^{ \star }}\right) = O\left( {\lambda {\left( \mathbf{K}\ri... | Proof. One notices that for any \( \left( {\mathbf{v},{\mathbf{v}}^{\prime }}\right) \in {\left( {\mathbb{R}}_{+, * }^{m}\right) }^{2} \), one has\n\n\[ \n{d}_{\mathcal{H}}\left( {\mathbf{v},{\mathbf{v}}^{\prime }}\right) = {d}_{\mathcal{H}}\left( {\mathbf{v}/{\mathbf{v}}^{\prime },{\mathbb{1}}_{m}}\right) = {d}_{\math... | Yes |
Proposition 4.4. One has\n\n\[ \n{\mathrm{L}}_{\mathbf{C}}^{\varepsilon }\left( {\mathbf{a},\mathbf{b}}\right) = \mathop{\max }\limits_{{\mathbf{f} \in {\mathbb{R}}^{n},\mathbf{g} \in {\mathbb{R}}^{m}}}\langle \mathbf{f},\mathbf{a}\rangle + \langle \mathbf{g},\mathbf{b}\rangle - \varepsilon \left\langle {{e}^{\mathbf{f... | Proof. We start from the end of the proof of Proposition 4.3, which links the optimal primal solution \( \mathbf{P} \) and dual multipliers \( \mathbf{f} \) and \( \mathbf{g} \) for the marginal constraints as\n\n\[ \n{\mathbf{P}}_{i, j} = {e}^{{\mathbf{f}}_{i}/\varepsilon }{e}^{-{\mathbf{C}}_{i, j}/\varepsilon }{e}^{{... | Yes |
Proposition 4.5. Any pair of optimal solutions \( \left( {{\mathbf{f}}^{ \star },{\mathbf{g}}^{ \star }}\right) \) to (4.30) are such that \( \left( {{\mathbf{f}}^{ \star },{\mathbf{g}}^{ \star }}\right) \in \) \( \mathbf{R}\left( \mathbf{C}\right) \), the set of feasible Kantorovich potentials defined in (2.21). As a ... | Proof. Primal-dual optimality conditions in (4.4) with the constraint that \( \mathbf{P} \) is a probability and therefore \( {\mathbf{P}}_{i, j} \leq 1 \) for all \( i, j \) yields that \( \exp \left( {-\left( {{\mathbf{f}}_{i}^{ \star } + {\mathbf{g}}_{j}^{ \star } - {\mathbf{C}}_{i, j}}\right) /\varepsilon }\right) ... | Yes |
Proposition 4.6. \( {\mathrm{L}}_{\mathbf{C}}^{\varepsilon }\left( {\mathbf{a},\mathbf{b}}\right) \) is a jointly convex function of \( \mathbf{a} \) and \( \mathbf{b} \) for \( \varepsilon \geq 0 \) . When \( \varepsilon > 0 \), its gradient is equal to | \[ \nabla {L}_{\mathbf{C}}^{\varepsilon }\left( {\mathbf{a},\mathbf{b}}\right) = \left\lbrack \begin{matrix} {\mathbf{f}}^{ \star } \\ {\mathbf{g}}^{ \star } \end{matrix}\right\rbrack \] where \( {\mathbf{f}}^{ \star } \) and \( {\mathbf{g}}^{ \star } \) are the optimal solutions of Equation (4.30) chosen so that their... | Yes |
Proposition 4.7. The following relationship holds:\n\n\[ \n{\mathfrak{D}}_{\mathbf{C}}^{\varepsilon }\left( {\mathbf{a},\mathbf{b}}\right) \leq {\mathrm{L}}_{\mathbf{C}}^{\varepsilon }\left( {\mathbf{a},\mathbf{b}}\right) \leq {\mathfrak{P}}_{\mathbf{C}}^{\varepsilon }\left( {\mathbf{a},\mathbf{b}}\right) \n\]\n\nFurth... | Proof. Equation (4.47) is obtained by writing that the primal and dual problems have the same values at the optima (see (4.30)), and hence\n\n\[ \n{\mathcal{L}}_{\mathbf{C}}^{\varepsilon }\left( {\mathbf{a},\mathbf{b}}\right) = {\mathfrak{P}}_{\mathbf{C}}^{\varepsilon }\left( {\mathbf{a},\mathbf{b}}\right) - \varepsilo... | Yes |
Proposition 4.8 (Finite Sinkhorn divergences). The following relationship holds:\n\n\[ \n{\mathfrak{D}}_{\mathbf{C}}^{\left( L\right) }\left( {\mathbf{a},\mathbf{b}}\right) \leq {\mathrm{L}}_{\mathbf{C}}^{\varepsilon }\left( {\mathbf{a},\mathbf{b}}\right) \n\] | Proof. Similarly to the proof of Proposition 4.5, we exploit the fact that after even just one single Sinkhorn iteration, we have, following (4.35) and (4.36), that \( {\mathbf{f}}^{\left( L\right) } \) and \( {\mathbf{g}}^{\left( L\right) } \) are such that the matrix with elements \( \exp \left( {-\left( {{\mathbf{f}... | Yes |
Proposition 6.1. Suppose \( \mathcal{X} = \mathcal{Y} \) and \( c\left( {x, y}\right) = d\left( {x, y}\right) \) . Then, there exists \( g \) such that \( f = {g}^{c} \) if and only \( \operatorname{Lip}\left( f\right) \leq 1 \) . Furthermore, if \( \operatorname{Lip}\left( f\right) \leq 1 \), then \( {f}^{c} = - f \) ... | Proof. First, suppose \( f = {g}^{c} \) . Then, for \( x, y \in \mathcal{X} \) ,\n\n\[ \left| {f\left( x\right) - f\left( y\right) }\right| = \left| {\mathop{\inf }\limits_{{z \in \mathcal{X}}}d\left( {x, z}\right) - g\left( z\right) - \mathop{\inf }\limits_{{z \in \mathcal{X}}}d\left( {y, z}\right) - g\left( z\right) ... | Yes |
Example 8.1 (Kullback-Leibler divergence). The Kullback-Leibler divergence KL \( \overset{\underset{\mathrm{{def}.}}{}}{ = } \) \( {\mathcal{D}}_{{\varphi }_{\mathrm{{KL}}}} \), also known as the relative entropy, was already introduced in (4.10) and (4.6). It is the divergence associated to the Shannon-Boltzman entrop... | \[ {\varphi }_{\mathrm{{KL}}}\left( s\right) = \left\{ \begin{array}{ll} s\log \left( s\right) - s + 1 & \text{ for }s > 0 \\ 1 & \text{ for }s = 0 \\ + \infty & \text{ otherwise. } \end{array}\right. \] | Yes |
Example 8.3 (Hellinger). The Hellinger distance \( \mathfrak{h}\overset{\text{ def. }}{ = }{\mathcal{D}}_{{\varphi }_{H}}^{1/2} \) is the square root of the divergence associated to | \[ {\varphi }_{H}\left( s\right) = \left\{ \begin{array}{ll} {\left| \sqrt{s} - 1\right| }^{2} & \text{ for }s \geq 0 \\ + \infty & \text{ otherwise. } \end{array}\right. \] As its name suggests, \( \mathfrak{h} \) is a distance on \( {\mathcal{M}}_{ + }\left( \mathcal{X}\right) \), which metrizes the strong topology a... | Yes |
The Jensen-Shannon distance \( \operatorname{JS}\left( {\alpha ,\beta }\right) \), defined as \[ \operatorname{JS}{\left( \alpha ,\beta \right) }^{2}\overset{\text{ def. }}{ = }\frac{1}{2}\left( {\operatorname{KL}\left( {\alpha \mid \xi }\right) + \operatorname{KL}\left( {\beta \mid \xi }\right) }\right) \;\text{ where... | is a distance [Endres and Schindelin,2003,Österreicher and Vajda,2003]. \( {\mathrm{{JS}}}^{2} \) can be shown to be a \( \varphi \) -divergence for \( \varphi \left( s\right) = t\log \left( t\right) - \left( {t + 1}\right) \log \left( {t + 1}\right) \) . In sharp contrast with \( \mathrm{{KL}},\mathrm{{JS}}\left( {\al... | No |
Another important dual norm is \( {H}^{-1}\left( {\mathbb{R}}^{d}\right) \), the dual (over distributions) of the Sobolev space \( {H}^{1}\left( {\mathbb{R}}^{d}\right) \) of functions having derivatives in \( {L}^{2}\left( {\mathbb{R}}^{d}\right) \) . It is defined using the primal RKHS norm \( \parallel \nabla f{\par... | \[ \parallel \alpha - \beta {\parallel }_{{H}^{-1}\left( {\mathbb{R}}^{d}\right) }^{2} = \mathop{\min }\limits_{s}\left\{ {{\int }_{{\mathbb{R}}^{d}}\parallel s\left( x\right) {\parallel }_{2}^{2}\mathrm{\;d}x : \operatorname{div}\left( s\right) = \alpha - \beta }}\right\} ,\] | Yes |
One can generalize this construction by considering the Sobolev space \( {H}^{-r}\left( {\mathbb{R}}^{d}\right) \) of arbitrary negative index, which is the dual of the functional Sobolev space \( {H}^{r}\left( {\mathbb{R}}^{d}\right) \) of functions having \( r \) derivatives (in the sense of distributions) in \( {L}^... | In order to metrize the weak convergence, one needs functions in \( {H}^{r}\left( {\mathbb{R}}^{d}\right) \) to be continuous, which is the case when \( r > d/2 \) . As the dimension \( d \) increases, one thus needs to consider higher regularity. For arbitrary \( \alpha \) (not necessarily integers), these spaces are ... | Yes |
Example 8.12 (Energy distance). The energy distance (or Cramer distance when \( d = \) 1) [Székely and Rizzo,2004] associated to a distance \( d \) is defined as\n\n\[ \parallel \alpha - \beta {\parallel }_{\mathrm{{ED}}\left( {\mathcal{X},{d}^{p}}\right) }\overset{\text{ def. }}{ = }\parallel \alpha - \beta {\parallel... | For \( \mathcal{X} = {\mathbb{R}}^{d}, d\left( {x, y}\right) = \parallel \cdot \parallel \), using (8.18), one sees that the energy distance is a Sobolev norm\n\n\[ \parallel \cdot {\parallel }_{\mathrm{{ED}}\left( {{\mathbb{R}}^{d},\parallel \cdot {\parallel }^{p}}\right) } = \parallel \cdot {\parallel }_{{H}^{-\frac{... | Yes |
Proposition 8.1. A distance \( d \) is Hilbertian if and only if \( {d}^{2} \) is negative definite. | Proof. If a distance is Hilbertian, then \( {d}^{2} \) is trivially negative definite. Indeed, given \( n \) points in \( \mathcal{Z} \), the sum \( \sum {r}_{i}{r}_{j}{d}^{2}\left( {{z}_{i},{z}_{j}}\right) \) can be rewritten as \( \sum {r}_{i}{r}_{j}{\begin{Vmatrix}\phi \left( {z}_{i}\right) - \phi \left( {z}_{j}\rig... | Yes |
Proposition 8.2. If \( \mathcal{X} = {\mathbb{R}}^{d} \) with \( d \geq 2 \) and the ground cost is set to \( d\left( {x, y}\right) = \parallel x - y{\parallel }_{2} \) , then the \( p \) -Wasserstein distance is not Hilbertian for \( p = 1,2 \) . | Proof. It suffices to prove the result for \( d = 2 \) since any counterexample in that dimension suffices to obtain a counterexample in any higher dimension. We provide a nonrandom counterexample which works using measures supported on four vectors \( {x}^{1},{x}^{2},{x}^{3},{x}^{4} \in {\mathbb{R}}^{2} \) defined as ... | Yes |
Proposition 8.3. One has\n\n\[ \n{\widetilde{\mathcal{W}}}_{p,\varepsilon }\left( {\alpha ,\beta }\right) \overset{\varepsilon \rightarrow 0}{ \rightarrow }2{\mathcal{W}}_{p}\left( {\alpha ,\beta }\right) \;\text{ and }\;{\widetilde{\mathcal{W}}}_{p,\varepsilon }{\left( \alpha ,\beta \right) }^{p}\overset{\varepsilon \... | Figure 8.9 shows numerically the impact of \( \varepsilon \) on the sample complexity rates. It is proved in Genevay et al. [2019], in the case of \( c\left( {x, y}\right) = \parallel x - y{\parallel }^{2} \) on \( \mathcal{X} = {\mathbb{R}}^{d} \), that these rates interpolate between the ones of OT and MMD. | No |
Proposition 9.1 (Derivative with respect to histograms). For \( \varepsilon > 0,\left( {\mathbf{a},\mathbf{b}}\right) \mapsto {\mathrm{L}}_{\mathbf{C}}^{\varepsilon }\left( {\mathbf{a},\mathbf{b}}\right) \) is convex and differentiable. Its gradient reads | \[ \nabla {\mathrm{L}}_{\mathbf{C}}^{\varepsilon }\left( {\mathbf{a},\mathbf{b}}\right) = \left( {\mathbf{f},\mathbf{g}}\right) \] where \( \left( {\mathbf{f},\mathbf{g}}\right) \) is the unique solution to (4.30), centered such that \( \mathop{\sum }\limits_{i}{\mathbf{f}}_{i} = \mathop{\sum }\limits_{j}{\mathbf{g}}_{... | Yes |
Proposition 9.2 (Derivative with respect to the cost). For fixed input histograms \( \left( {\mathbf{a},\mathbf{b}}\right) \) , for \( \varepsilon > 0 \), the mapping \( \mathbf{C} \mapsto \mathcal{R}\left( \mathbf{C}\right) \overset{\text{ def }}{ = }{\mathrm{L}}_{\mathbf{C}}^{\varepsilon }\left( {\mathbf{a},\mathbf{b... | \[ \nabla \mathcal{R}\left( \mathbf{C}\right) = \mathbf{P} \] (9.5) where \( \mathbf{P} \) is the unique optimal solution of (4.2). For \( \varepsilon = 0 \), this formula defines the set of upper gradients. | Yes |
Proposition 9.1. The optimal \( \left( {{\mathbf{u}}_{s},{\mathbf{v}}_{s}}\right) \) appearing in (9.17) can be written as \( \left( {{\mathbf{u}}_{s},{\mathbf{v}}_{s}}\right) = \) \( \left( {{e}^{{\mathbf{f}}_{s}/\varepsilon },{e}^{{\mathbf{g}}_{s}/\varepsilon }}\right) \), where \( {\left( {\mathbf{f}}_{s},{\mathbf{g... | Proof. Introducing Lagrange multipliers in (9.16) leads to\n\n\[ \mathop{\min }\limits_{{{\left( {\mathbf{P}}_{s}\right) }_{s},\mathbf{a}}}\mathop{\max }\limits_{{\left( {\mathbf{f}}_{s},{\mathbf{g}}_{s}\right) }_{s}}\mathop{\sum }\limits_{s}{\lambda }_{s}\left( {\varepsilon \mathbf{{KL}}\left( {{\mathbf{P}}_{s} \mid {... | Yes |
Example 2.3 The second-order cone is the norm cone for the Euclidean norm, i.e., | \[ C = \left\{ {\left( {x, t}\right) \in {\mathbf{R}}^{n + 1} \mid \parallel x{\parallel }_{2} \leq t}\right\} \] \[ = \left\{ {\left. \left\lbrack \begin{array}{l} x \\ t \end{array}\right\rbrack \right| \;{\left\lbrack \begin{array}{l} x \\ t \end{array}\right\rbrack }^{T}\left\lbrack \begin{matrix} I & 0 \\ 0 & - 1 ... | Yes |
To describe the simplex (2.7) as a polyhedron, i.e., in the form (2.6), we proceed as follows. By definition, \( x \in C \) if and only if \( x = {\theta }_{0}{v}_{0} + {\theta }_{1}{v}_{1} + \cdots + {\theta }_{k}{v}_{k} \) for some \( \theta \succcurlyeq 0 \) with \( {\mathbf{1}}^{T}\theta = 1 \) . Equivalently, if w... | \[ B = \left\lbrack \begin{array}{lll} {v}_{1} - {v}_{0} & \cdots & {v}_{k} - {v}_{0} \end{array}\right\rbrack \in {\mathbf{R}}^{n \times k}, \] we can say that \( x \in C \) if and only if \[ x = {v}_{0} + {By} \] (2.8) for some \( y \succcurlyeq 0 \) with \( {\mathbf{1}}^{T}y \leq 1 \) . Now we note that affine indep... | Yes |
Example 2.7 The positive semidefinite cone \( {\mathbf{S}}_{ + }^{n} \) can be expressed as\n\n\[ \mathop{\bigcap }\limits_{{z \neq 0}}\left\{ {X \in {\mathbf{S}}^{n} \mid {z}^{T}{Xz} \geq 0}\right\} \] | For each \( z \neq 0,{z}^{T}{Xz} \) is a (not identically zero) linear function of \( X \), so the sets\n\n\[ \left\{ {X \in {\mathbf{S}}^{n} \mid {z}^{T}{Xz} \geq 0}\right\} \]\n\nare, in fact, halfspaces in \( {\mathbf{S}}^{n} \) . Thus the positive semidefinite cone is the intersection of an infinite number of halfs... | Yes |
Example 2.8 We consider the set\n\n\[ S = \\left\\{ {x \\in {\\mathbf{R}}^{m}\\left| \\right| p\\left( t\\right) \\mid \\leq 1\\text{ for }\\left| t\\right| \\leq \\pi /3}\\right\\} \]\n\nwhere \( p\\left( t\\right) = \\mathop{\\sum }\\limits_{{k = 1}}^{m}{x}_{k}\\cos {kt} \) . The set \( S \) can be expressed as the i... | and so is convex. | No |
Example 2.9 Polyhedron. The polyhedron \( \{ x \mid {Ax} \preccurlyeq b,{Cx} = d\} \) can be expressed as the inverse image of the Cartesian product of the nonnegative orthant and the origin under the affine function \( f\left( x\right) = \left( {b - {Ax}, d - {Cx}}\right) \) : | \[ \{ x \mid {Ax} \preccurlyeq b,{Cx} = d\} = \left\{ {x \mid f\left( x\right) \in {\mathbf{R}}_{ + }^{m}\times \{ 0\} }\right\} . \] | Yes |
The solution set of a linear matrix inequality, \( \{ x \mid A\left( x\right) \preccurlyeq B\} \), is convex. | Indeed, it is the inverse image of the positive semidefinite cone under the affine function \( f : {\mathbf{R}}^{n} \rightarrow {\mathbf{S}}^{m} \) given by \( f\left( x\right) = B - A\left( x\right) \). | Yes |
Example 2.11 Hyperbolic cone. The set\n\n\[ \left\{ {x \mid {x}^{T}{Px} \leq {\left( {c}^{T}x\right) }^{2},{c}^{T}x \geq 0}\right\} \]\n\nwhere \( P \in {\mathbf{S}}_{ + }^{n} \) and \( c \in {\mathbf{R}}^{n} \), is convex, since it is the inverse image of the second-order cone,\n\n\[ \left\{ {\left( {z, t}\right) \mid... | since it is the inverse image of the second-order cone,\n\n\[ \left\{ {\left( {z, t}\right) \mid {z}^{T}z \leq {t}^{2}, t \geq 0}\right\} \]\n\nunder the affine function \( f\left( x\right) = \left( {{P}^{1/2}x,{c}^{T}x}\right) \) | Yes |
Suppose \( u \) and \( v \) are random variables that take on values in \( \{ 1,\ldots, n\} \) and \( \{ 1,\ldots, m\} \), respectively, and let \( {p}_{ij} \) denote \( \operatorname{prob}\left( {u = i, v = j}\right) \) . Then the conditional probability \( {f}_{ij} = \operatorname{prob}\left( {u = i \mid v = j}\right... | \[ {f}_{ij} = \frac{{p}_{ij}}{\mathop{\sum }\limits_{{k = 1}}^{n}{p}_{kj}}. \] | Yes |
Example 2.16 Cone of polynomials nonnegative on \( \left\lbrack {0,1}\right\rbrack \) . Let \( K \) be defined as\n\n\[ K = \left\{ {c \in {\mathbf{R}}^{n} \mid {c}_{1} + {c}_{2}t + \cdots + {c}_{n}{t}^{n - 1} \geq 0\text{ for }t \in \left\lbrack {0,1}\right\rbrack }\right\} ,\]\n\n\( \left( {2.15}\right) \)\n\ni.e., \... | Two vectors \( c, d \in {\mathbf{R}}^{n} \) satisfy \( c{ \preccurlyeq }_{K}d \) if and only if\n\n\[ {c}_{1} + {c}_{2}t + \cdots + {c}_{n}{t}^{n - 1} \leq {d}_{1} + {d}_{2}t + \cdots + {d}_{n}{t}^{n - 1} \]\n\nfor all \( t \in \left\lbrack {0,1}\right\rbrack \) . | Yes |
Example 2.19 Separation of an affine and a convex set. Suppose \( C \) is convex and \( D \) is affine, i.e., \( D = \left\{ {{Fu} + g \mid u \in {\mathbf{R}}^{m}}\right\} \), where \( F \in {\mathbf{R}}^{n \times m} \). Suppose \( C \) and \( D \) are disjoint, so by the separating hyperplane theorem there are \( a \n... | Now \( {a}^{T}x \geq b \) for all \( x \in D \) means \( {a}^{T}{Fu} \geq b - {a}^{T}g \) for all \( u \in {\mathbf{R}}^{m} \). But a linear function is bounded below on \( {\mathbf{R}}^{m} \) only when it is zero, so we conclude \( {a}^{T}F = 0 \) (and hence, \( b \leq {a}^{T}g \) ).\n\nThus we conclude that there exi... | Yes |
Strict separation of a point and a closed convex set. Let \( C \) be a closed convex set and \( {x}_{0} \notin C \) . Then there exists a hyperplane that strictly separates \( {x}_{0} \) from \( C \) . | To see this, note that the two sets \( C \) and \( B\left( {{x}_{0},\epsilon }\right) \) do not intersect for some \( \epsilon > 0 \) . By the separating hyperplane theorem, there exist \( a \neq 0 \) and \( b \) such that \( {a}^{T}x \leq b \) for \( x \in C \) and \( {a}^{T}x \geq b \) for \( x \in B\left( {{x}_{0},\... | Yes |
We derive the necessary and sufficient conditions for solvability of a system of strict linear inequalities\n\n\[ \n{Ax} \prec b\text{.}\n\]\n\n\( \left( {2.17}\right) \) | These inequalities are infeasible if and only if the (convex) sets\n\n\[ \nC = \left\{ {b - {Ax} \mid x \in {\mathbf{R}}^{n}}\right\} ,\;D = {\mathbf{R}}_{+ + }^{m} = \left\{ {y \in {\mathbf{R}}^{m} \mid y \succ 0}\right\}\n\]\n\ndo not intersect. The set \( D \) is open; \( C \) is an affine set. Hence by the result a... | Yes |
Example 2.23 Nonnegative orthant. The cone \( {\mathbf{R}}_{ + }^{n} \) is its own dual: | \[ {x}^{T}y \geq 0\text{ for all }x \succcurlyeq 0 \Leftrightarrow y \succcurlyeq 0. \] | Yes |
Positive semidefinite cone. On the set of symmetric \( n \times n \) matrices \( {\mathbf{S}}^{n} \), we use the standard inner product \( \operatorname{tr}\left( {XY}\right) = \mathop{\sum }\limits_{{i, j = 1}}^{n}{X}_{ij}{Y}_{ij} \) (see §A.1.1). The positive semidefinite cone \( {\mathbf{S}}_{ + }^{n} \) is self-dua... | We will establish this fact. Suppose \( Y \notin {\mathbf{S}}_{ + }^{n} \) . Then there exists \( q \in {\mathbf{R}}^{n} \) with \[ {q}^{T}{Yq} = \operatorname{tr}\left( {q{q}^{T}Y}\right) < 0. \] Hence the positive semidefinite matrix \( X = q{q}^{T} \) satisfies \( \operatorname{tr}\left( {XY}\right) < 0 \) ; it foll... | Yes |
Example 2.25 Dual of a norm cone. Let \( \parallel \cdot \parallel \) be a norm on \( {\mathbf{R}}^{n} \) . The dual of the associated cone \( K = \left\{ {\left( {x, t}\right) \in {\mathbf{R}}^{n + 1} \mid \parallel x\parallel \leq t}\right\} \) is the cone defined by the dual norm, i.e., \[ {K}^{ * } = \left\{ {\left... | To prove the result we have to show that \[ {x}^{T}u + {tv} \geq 0\text{ whenever }\parallel x\parallel \leq t \Leftrightarrow \parallel u{\parallel }_{ * } \leq v. \] Let us start by showing that the righthand condition on \( \left( {u, v}\right) \) implies the lefthand condition. Suppose \( \parallel u{\parallel }_{ ... | Yes |
Example 2.26 Theorem of alternatives for linear strict generalized inequalities. Suppose \( K \subseteq {\mathbf{R}}^{m} \) is a proper cone. Consider the strict generalized inequality\n\n\[ \n{Ax}{ \prec }_{K}b \n\]\n\n\( \left( {2.21}\right) \)\n\nwhere \( x \in {\mathbf{R}}^{n} \) . | We will derive a theorem of alternatives for this inequality. Suppose it is infeasible, i.e., the affine set \( \left\{ {b - {Ax} \mid x \in {\mathbf{R}}^{n}}\right\} \) does not intersect the open convex set int \( K \) . Then there is a separating hyperplane, i.e., a nonzero \( \lambda \in {\mathbf{R}}^{m} \) and \( ... | Yes |
Example 2.27 Pareto optimal production frontier. We consider a product which requires \( n \) resources (such as labor, electricity, natural gas, water) to manufacture. The product can be manufactured or produced in many ways. With each production method, we associate a resource vector \( x \in {\mathbf{R}}^{n} \), whe... | The production set \( P \subseteq {\mathbf{R}}^{n} \) is defined as the set of all resource vectors \( x \) that correspond to some production method.\n\nProduction methods with resource vectors that are minimal elements of \( P \), with respect to componentwise inequality, are called Pareto optimal or efficient. The s... | Yes |
Example 3.1 Indicator function of a convex set. Let \( C \subseteq {\mathbf{R}}^{n} \) be a convex set, and consider the (convex) function \( {I}_{C} \) with domain \( C \) and \( {I}_{C}\left( x\right) = 0 \) for all \( x \in C \) . In other words, the function is identically zero on the set \( C \) . Its extended-val... | is given by\n\n\[ \n{\widetilde{I}}_{C}\left( x\right) = \left\{ \begin{array}{ll} 0 & x \in C \\ \infty & x \notin C. \end{array}\right. \n\]\n\nThe convex function \( {\widetilde{I}}_{C} \) is called the indicator function of the set \( C \) . | Yes |
Consider the quadratic function \( f : {\mathbf{R}}^{n} \rightarrow \mathbf{R} \), with dom \( f = {\mathbf{R}}^{n} \), given by \[ f\left( x\right) = \left( {1/2}\right) {x}^{T}{Px} + {q}^{T}x + r, \] with \( P \in {\mathbf{S}}^{n}, q \in {\mathbf{R}}^{n} \), and \( r \in \mathbf{R} \). | Since \( {\nabla }^{2}f\left( x\right) = P \) for all \( x, f \) is convex if and only if \( P \succcurlyeq 0 \) (and concave if and only if \( P \preccurlyeq 0 \). For quadratic functions, strict convexity is easily characterized: \( f \) is strictly convex if and only if \( P \succ 0 \) (and strictly concave if and o... | Yes |
The function \( f : {\mathbf{R}}^{n} \times {\mathbf{S}}^{n} \rightarrow \mathbf{R} \), defined as\n\n\[ f\left( {x, Y}\right) = {x}^{T}{Y}^{-1}x \]\n\nis convex on \( \operatorname{dom}f = {\mathbf{R}}^{n} \times {\mathbf{S}}_{+ + }^{n} \). | One easy way to establish convexity of \( f \) is via its epigraph:\n\n\[ \text{ epi }f = \left\{ {\left( {x, Y, t}\right) \mid Y \succ 0,{x}^{T}{Y}^{-1}x \leq t}\right\} \]\n\n\[ = \left\{ {\left( {x, Y, t}\right) \left| {\;\left\lbrack \begin{matrix} Y & x \\ {x}^{T} & t \end{matrix}\right\rbrack \succcurlyeq 0}\righ... | Yes |
The function\n\n\[ f\left( x\right) = \max \left\{ {{a}_{1}^{T}x + {b}_{1},\ldots ,{a}_{L}^{T}x + {b}_{L}}\right\} \]\n\ndefines a piecewise-linear (or really, affine) function (with \( L \) or fewer regions). It is convex since it is the pointwise maximum of affine functions. | The converse can also be shown: any piecewise-linear convex function with \( L \) or fewer regions can be expressed in this form. (See exercise 3.29.) | No |
For \( x \in {\mathbf{R}}^{n} \) we denote by \( {x}_{\left\lbrack i\right\rbrack } \) the \( i \) th largest component of \( x \), i.e., \[ {x}_{\left\lbrack 1\right\rbrack } \geq {x}_{\left\lbrack 2\right\rbrack } \geq \cdots \geq {x}_{\left\lbrack n\right\rbrack } \] are the components of \( x \) sorted in nonincrea... | This can be seen by writing it as \[ f\left( x\right) = \mathop{\sum }\limits_{{i = 1}}^{r}{x}_{\left\lbrack i\right\rbrack } = \max \left\{ {{x}_{{i}_{1}} + \cdots + {x}_{{i}_{r}} \mid 1 \leq {i}_{1} < {i}_{2} < \cdots < {i}_{r} \leq n}\right\} , \] i.e., the maximum of all possible sums of \( r \) different component... | Yes |
Example 3.7 Support function of a set. Let \( C \subseteq {\mathbf{R}}^{n} \), with \( C \neq \varnothing \) . The support function \( {S}_{C} \) associated with the set \( C \) is defined as\n\n\[ \n{S}_{C}\left( x\right) = \sup \left\{ {{x}^{T}y \mid y \in C}\right\} \n\]\n\n(and, naturally, \( \left. {\operatorname{... | For each \( y \in C,{x}^{T}y \) is a linear function of \( x \), so \( {S}_{C} \) is the pointwise supremum of a family of linear functions, hence convex. | Yes |
Example 3.8 Distance to farthest point of a set. Let \( C \subseteq {\mathbf{R}}^{n} \) . The distance (in any norm) to the farthest point of \( C \) , \[ f\left( x\right) = \mathop{\sup }\limits_{{y \in C}}\parallel x - y\parallel \] | is convex. To see this, note that for any \( y \), the function \( \parallel x - y\parallel \) is convex in \( x \) . Since \( f \) is the pointwise supremum of a family of convex functions (indexed by \( y \in C \) ), it is a convex function of \( x \) . | Yes |
Let \( {a}_{1},\ldots ,{a}_{n} \in {\mathbf{R}}^{m} \). In a weighted least-squares problem we minimize the objective function \( \mathop{\sum }\limits_{{i = 1}}^{n}{w}_{i}\left( {{a}_{i}^{T}x - }\right. \) \( {\left. {b}_{i}\right) }^{2} \) over \( x \in {\mathbf{R}}^{m} \). We refer to \( {w}_{i} \) as weights, and a... | Since \( g \) is the infimum of a family of linear functions of \( w \) (indexed by \( x \in {\mathbf{R}}^{m} \) ), it is a concave function of \( w \). We can derive an explicit expression for \( g \), at least on part of its domain. Let \( W = \operatorname{diag}\left( w\right) \), the diagonal matrix with elements \... | Yes |
Example 3.10 Maximum eigenvalue of a symmetric matrix. The function \( f\left( X\right) = \) \( {\lambda }_{\max }\left( X\right) \), with \( \operatorname{dom}f = {\mathbf{S}}^{m} \), is convex. | To see this, we express \( f \) as\n\n\[ f\left( X\right) = \sup \left\{ {{y}^{T}{Xy} \mid \parallel y{\parallel }_{2} = 1}\right\} \]\n\ni.e., as the pointwise supremum of a family of linear functions of \( X \) (i.e., \( {y}^{T}{Xy} \) ) indexed by \( y \in {\mathbf{R}}^{m} \) . | Yes |
Consider \( f\left( X\right) = \parallel X{\parallel }_{2} \) with \( \operatorname{dom}f = {\mathbf{R}}^{p \times q} \) , where \( \parallel \cdot {\parallel }_{2} \) denotes the spectral norm or maximum singular value. Convexity of \( f \) follows from | \[ f\left( X\right) = \sup \left\{ {{u}^{T}{Xv} \mid \parallel u{\parallel }_{2} = 1,\parallel v{\parallel }_{2} = 1}\right\} ,\] which shows it is the pointwise supremum of a family of linear functions of \( X \) . | Yes |
if \( g \) is convex, \( h \) is convex, and \( \widetilde{h} \) is nondecreasing, then \( f = h \circ g \) is convex. | Assume that \( x, y \in \operatorname{dom}f \), and \( 0 \leq \theta \leq 1 \) . Since \( x, y \in \operatorname{dom}f \), we have that \( x, y \in \operatorname{dom}g \) and \( g\left( x\right), g\left( y\right) \in \operatorname{dom}h \) . Since \( \operatorname{dom}g \) is convex, we conclude that \( {\theta x} + \l... | Yes |
Suppose \( p \geq 1 \), and \( {g}_{1},\ldots ,{g}_{k} \) are convex and nonnegative. Then the function \( {\left( \mathop{\sum }\limits_{{i = 1}}^{k}{g}_{i}{\left( x\right) }^{p}\right) }^{1/p} \) is convex. | To show this, we consider the function \( h : {\mathbf{R}}^{k} \rightarrow \mathbf{R} \) defined as\n\n\[ h\left( z\right) = {\left( \mathop{\sum }\limits_{{i = 1}}^{k}\max {\left\{ {z}_{i},0\right\} }^{p}\right) }^{1/p}, \]\n\nwith \( \operatorname{dom}h = {\mathbf{R}}^{k} \), so \( h = \widetilde{h} \) . This functio... | Yes |
Suppose the quadratic function\n\n\[ f\left( {x, y}\right) = {x}^{T}{Ax} + 2{x}^{T}{By} + {y}^{T}{Cy}, \]\n\n(where \( A \) and \( C \) are symmetric) is convex in \( \left( {x, y}\right) \), which means\n\n\[ \left\lbrack \begin{matrix} A & B \\ {B}^{T} & C \end{matrix}\right\rbrack \succcurlyeq 0 \]\n\nWe can express... | By the minimization rule, \( g \) is convex, so we conclude that \( A - B{C}^{ \dagger }{B}^{T} \succcurlyeq 0 \). | Yes |
Suppose \( h \) is convex. Then the function \( g \) defined as\n\n\[ g\left( x\right) = \inf \{ h\left( y\right) \mid {Ay} = x\} \]\n\nis convex. | To see this, we define \( f \) by\n\n\[ f\left( {x, y}\right) = \left\{ \begin{array}{ll} h\left( y\right) & \text{ if }{Ay} = x \\ \infty & \text{ otherwise,} \end{array}\right. \]\n\nwhich is convex in \( \left( {x, y}\right) \) . Then \( g \) is the minimum of \( f \) over \( y \), and hence is convex. (It is not ha... | Yes |
Example 3.18 Euclidean norm squared. The perspective of the convex function \( f\left( x\right) = {x}^{T}x \) on \( {\mathbf{R}}^{n} \) is | \[ g\left( {x, t}\right) = t{\left( x/t\right) }^{T}\left( {x/t}\right) = \frac{{x}^{T}x}{t}, \] which is convex in \( \left( {x, t}\right) \) for \( t > 0 \) . We can deduce convexity of \( g \) using several other methods. First, we can express \( g \) as the sum of the quadratic-over-linear functions \( {x}_{i}^{2}/... | Yes |
We derive the conjugates of some convex functions on \( \mathbf{R} \) . | - Affine function. \( f\left( x\right) = {ax} + b \) . As a function of \( x,{yx} - {ax} - b \) is bounded if and only if \( y = a \), in which case it is constant. Therefore the domain of the conjugate function \( {f}^{ * } \) is the singleton \( \{ a\} \), and \( {f}^{ * }\left( a\right) = - b \) .\n\n- Negative loga... | Yes |
Consider \( f\left( x\right) = \frac{1}{2}{x}^{T}{Qx} \), with \( Q \in {\mathbf{S}}_{+ + }^{n} \) . The function \( {y}^{T}x - \frac{1}{2}{x}^{T}{Qx} \) is bounded above as a function of \( x \) for all \( y \) . | It attains its maximum at \( x = {Q}^{-1}y \), so\n\n\[ \n{f}^{ * }\left( y\right) = \frac{1}{2}{y}^{T}{Q}^{-1}y \n\] | Yes |
We consider \( f\left( X\right) = \log \det {X}^{-1} \) on \( {\mathbf{S}}_{+ + }^{n} \) . The conjugate function is defined as \[ {f}^{ * }\left( Y\right) = \mathop{\sup }\limits_{{X \succ 0}}\left( {\operatorname{tr}\left( {YX}\right) + \log \det X}\right) \] | since \( \operatorname{tr}\left( {YX}\right) \) is the standard inner product on \( {\mathbf{S}}^{n} \) . We first show that \( \operatorname{tr}\left( {YX}\right) + \log \det X \) is unbounded above unless \( Y \prec 0 \) . If \( Y \nprec 0 \), then \( Y \) has an eigenvector \( v \) , with \( \parallel v{\parallel }_... | Yes |
Let \( {I}_{S} \) be the indicator function of a (not necessarily convex) set \( S \subseteq {\mathbf{R}}^{n} \), i.e., \( {I}_{S}\left( x\right) = 0 \) on \( \operatorname{dom}{I}_{S} = S \) . Its conjugate is | \[ {I}_{S}^{ * }\left( y\right) = \mathop{\sup }\limits_{{x \in S}}{y}^{T}x \] which is the support function of the set \( S \) . | Yes |
To derive the conjugate of the log-sum-exp function \( f\left( x\right) = \log \left( {\mathop{\sum }\limits_{{i = 1}}^{n}{e}^{{x}_{i}}}\right) \) | we first determine the values of \( y \) for which the maximum over \( x \) of \( {y}^{T}x - f\left( x\right) \) is attained. By setting the gradient with respect to \( x \) equal to zero, we obtain the condition\n\n\[ \n{y}_{i} = \frac{{e}^{{x}_{i}}}{\mathop{\sum }\limits_{{j = 1}}^{n}{e}^{{x}_{j}}},\;i = 1,\ldots, n.... | Yes |
Let \( \parallel \cdot \parallel \) be a norm on \( {\mathbf{R}}^{n} \), with dual norm \( \parallel \cdot {\parallel }_{ * } \) . We will show that the conjugate of \( f\left( x\right) = \parallel x\parallel \) is\n\n\[ \n{f}^{ * }\left( y\right) = \left\{ \begin{array}{ll} 0 & \parallel y{\parallel }_{ * } \leq 1 \\ ... | If \( \parallel y{\parallel }_{ * } > 1 \), then by definition of the dual norm, there is a \( z \in {\mathbf{R}}^{n} \) with \( \parallel z\parallel \leq 1 \) and \( {y}^{T}z > 1 \) . Taking \( x = {tz} \) and letting \( t \rightarrow \infty \), we have\n\n\[ \n{y}^{T}x - \parallel x\parallel = t\left( {{y}^{T}z - \pa... | Yes |
Now consider the function \( f\left( x\right) = \left( {1/2}\right) \parallel x{\parallel }^{2} \), where \( \parallel \cdot \parallel \) is a norm, with dual norm \( \parallel \cdot {\parallel }_{ * } \) . We will show that its conjugate is \( {f}^{ * }\left( y\right) = \left( {1/2}\right) \parallel y{\parallel }_{ * ... | From \( {y}^{T}x \leq \parallel y{\parallel }_{ * }\parallel x\parallel \), we conclude\n\n\[ \n{y}^{T}x - \left( {1/2}\right) \parallel x{\parallel }^{2} \leq \parallel y{\parallel }_{ * }\parallel x\parallel - \left( {1/2}\right) \parallel x{\parallel }^{2} \n\]\n\nfor all \( x \) . The righthand side is a quadratic ... | Yes |
We consider a business or enterprise that consumes \( n \) resources and produces a product that can be sold. We let \( r = \left( {{r}_{1},\ldots ,{r}_{n}}\right) \) denote the vector of resource quantities consumed, and \( S\left( r\right) \) denote the sales revenue derived from the product produced (as a function o... | \[ M\left( p\right) = \mathop{\sup }\limits_{r}\left( {S\left( r\right) - {p}^{T}r}\right) \] The function \( M\left( p\right) \) gives the maximum profit attainable, as a function of the resource prices. In terms of conjugate functions, we can express \( M \) as \[ M\left( p\right) = {\left( -S\right) }^{ * }\left( {-... | Yes |
Example 3.30 Length of a vector. We define the length of \( x \in {\mathbf{R}}^{n} \) as the largest index of a nonzero component, i.e., \[ f\left( x\right) = \max \left\{ {i \mid {x}_{i} \neq 0}\right\} . \] (We define the length of the zero vector to be zero.) | This function is quasiconvex on \( {\mathbf{R}}^{n} \), since its sublevel sets are subspaces: \[ f\left( x\right) \leq \alpha \Leftrightarrow {x}_{i} = 0\text{ for }i = \lfloor \alpha \rfloor + 1,\ldots, n. \] | Yes |
Example 3.31 Consider \( f : {\mathbf{R}}^{2} \rightarrow \mathbf{R} \), with \( \operatorname{dom}f = {\mathbf{R}}_{ + }^{2} \) and \( f\left( {{x}_{1},{x}_{2}}\right) = {x}_{1}{x}_{2} \). This function is neither convex nor concave since its Hessian | \[ {\nabla }^{2}f\left( x\right) = \left\lbrack \begin{array}{ll} 0 & 1 \\ 1 & 0 \end{array}\right\rbrack \] is indefinite; it has one positive and one negative eigenvalue. The function \( f \) is quasiconcave, however, since the superlevel sets \[ \left\{ {x \in {\mathbf{R}}_{ + }^{2} \mid {x}_{1}{x}_{2} \geq \alpha }... | Yes |
Example 3.32 Linear-fractional function. The function\n\n\[ f\left( x\right) = \frac{{a}^{T}x + b}{{c}^{T}x + d} \]\n\nwith \( \operatorname{dom}f = \left\{ {x \mid {c}^{T}x + d > 0}\right\} \), is quasiconvex, and quasiconcave, i.e., quasilinear. Its \( \alpha \) -sublevel set is\n\n\[ {S}_{\alpha } = \left\{ {x \mid ... | which is convex, since it is the intersection of an open halfspace and a closed halfspace. (The same method can be used to show its superlevel sets are convex.) | Yes |
Suppose \( a, b \in {\mathbf{R}}^{n} \), and define\n\n\[ f\left( x\right) = \frac{\parallel x - a{\parallel }_{2}}{\parallel x - b{\parallel }_{2}} \]\n\ni.e., the ratio of the Euclidean distance to \( a \) to the distance to \( b \) . Then \( f \) is quasiconvex on the halfspace \( \left\{ {x \mid \parallel x - a{\pa... | To see this, we consider the \( \alpha \) -sublevel set of \( f \), with \( \alpha \leq 1 \) since \( f\left( x\right) \leq 1 \) on the halfspace \( \left\{ {x \mid \parallel x - a{\parallel }_{2} \leq \parallel x - b{\parallel }_{2}}\right\} \) . This sublevel set is the set of points satisfying\n\n\[ \parallel x - a{... | Yes |
Internal rate of return. Let \( x = \left( {{x}_{0},{x}_{1},\ldots ,{x}_{n}}\right) \) denote a cash flow sequence over \( n \) periods, where \( {x}_{i} > 0 \) means a payment to us in period \( i \), and \( {x}_{i} < 0 \) means a payment by us in period \( i \) . We define the present value of a cash flow, with inter... | Internal rate of return is a quasiconcave function of \( x \) (restricted to \( {x}_{0} < 0,{x}_{1} + \cdots + \) \( \left. {{x}_{n} > 0}\right) \) . To see this, we note that\n\n\[ \operatorname{IRR}\left( x\right) \geq R \Leftrightarrow \operatorname{PV}\left( {x, r}\right) > 0\text{ for }0 \leq r < R. \]\n\nThe left... | Yes |
Example 3.35 Cardinality of a nonnegative vector. The cardinality or size of a vector \( x \in {\mathbf{R}}^{n} \) is the number of nonzero components, and denoted \( \operatorname{card}\left( x\right) \) . The function card is quasiconcave on \( {\mathbf{R}}_{ + }^{n} \) (but not \( {\mathbf{R}}^{n} \) ). | This follows immediately from the modified Jensen inequality\n\n\[\n\operatorname{card}\left( {x + y}\right) \geq \min \{ \operatorname{card}\left( x\right) ,\operatorname{card}\left( y\right) \}\n\]\n\nwhich holds for \( x, y \succcurlyeq 0 \) . | Yes |
Example 3.36 Rank of positive semidefinite matrix. The function \( \operatorname{rank}X \) is quasi-concave on \( {\mathbf{S}}_{ + }^{n} \) . | This follows from the modified Jensen inequality (3.19), \n\n\[ \operatorname{rank}\left( {X + Y}\right) \geq \min \{ \operatorname{rank}X,\operatorname{rank}Y\} \]\n\nwhich holds for \( X, Y \in {\mathbf{S}}_{ + }^{n} \) . (This can be considered an extension of the previous example, since \( \operatorname{rank}\left(... | Yes |
The maximum generalized eigenvalue of a pair of symmetric matrices \( \left( {X, Y}\right) \), with \( Y \succ 0 \), is defined as\n\n\[ \n{\lambda }_{\max }\left( {X, Y}\right) = \mathop{\sup }\limits_{{u \neq 0}}\frac{{u}^{T}{Xu}}{{u}^{T}{Yu}} = \sup \{ \lambda \mid \det \left( {{\lambda Y} - X}\right) = 0\} .\n\]\n\... | To see this we consider the expression\n\n\[ \n{\lambda }_{\max }\left( {X, Y}\right) = \mathop{\sup }\limits_{{u \neq 0}}\frac{{u}^{T}{Xu}}{{u}^{T}{Yu}}.\n\]\n\nFor each \( u \neq 0 \), the function \( {u}^{T}{Xu}/{u}^{T}{Yu} \) is linear-fractional in \( \left( {X, Y}\right) \), hence a quasiconvex function of \( \le... | Yes |
Example 3.38 Convex over concave function. Suppose \( p \) is a convex function, \( q \) is a concave function, with \( p\left( x\right) \geq 0 \) and \( q\left( x\right) > 0 \) on a convex set \( C \) . Then the function \( f \) defined by \( f\left( x\right) = p\left( x\right) /q\left( x\right) \), on \( C \), is qua... | Here we have\n\n\[ f\left( x\right) \leq t \Leftrightarrow p\left( x\right) - {tq}\left( x\right) \leq 0, \]\n\nso we can take \( {\phi }_{t}\left( x\right) = p\left( x\right) - {tq}\left( x\right) \) for \( t \geq 0 \) . For each \( t,{\phi }_{t} \) is convex and for each \( x,{\phi }_{t}\left( x\right) \) is decreasi... | Yes |
Example 3.40 Log-concave density functions. Many common probability density functions are log-concave. Two examples are the multivariate normal distribution,\n\n\[ f\\left( x\\right) = \\frac{1}{\\sqrt{{\\left( 2\\pi \\right) }^{n}\\det \\sum }}{e}^{-\\frac{1}{2}{\\left( x - \\bar{x}\\right) }^{T}{\\sum }^{-1}\\left( {... | since\n\n\[ \\log f\\left( X\\right) = \\log a + \\frac{p - n - 1}{2}\\log \\det X - \\frac{1}{2}\\operatorname{tr}\\left( {{\\sum }^{-1}X}\\right) ,\n\]\n\nwhich is a concave function of \( X \) . | Yes |
Example 3.41 Laplace transform of a nonnegative function and the moment and cumulant generating functions. Suppose \( p : {\mathbf{R}}^{n} \rightarrow \mathbf{R} \) satisfies \( p\left( x\right) \geq 0 \) for all \( x \) . The Laplace transform of \( p \) ,\n\n\[ P\left( z\right) = \int p\left( x\right) {e}^{-{z}^{T}x}... | Now suppose \( p \) is a density, i.e., satisfies \( \int p\left( x\right) {dx} = 1 \) . The function \( M\left( z\right) = P\left( {-z}\right) \) is called the moment generating function of the density. It gets its name from the fact that the moments of the density can be found from the derivatives of the moment gener... | Yes |
Example 3.44 Volume of polyhedron. Let \( A \in {\mathbf{R}}^{m \times n} \) . Define\n\n\[ \n{P}_{u} = \left\{ {x \in {\mathbf{R}}^{n} \mid {Ax} \preccurlyeq u}\right\} \n\]\n\nThen its volume vol \( {P}_{u} \) is a log-concave function of \( u \) . | To prove this, note that the function\n\n\[ \n\Psi \left( {x, u}\right) = \left\{ \begin{array}{ll} 1 & {Ax} \preccurlyeq u \\ 0 & \text{ otherwise,} \end{array}\right. \n\]\n\nis log-concave. By the integration result, we conclude that\n\n\[ \n\int \Psi \left( {x, u}\right) {dx} = \operatorname{vol}{P}_{u} \n\]\n\nis ... | Yes |
Example 3.47 Convexity with respect to componentwise inequality. A function \( f \) : \( {\mathbf{R}}^{n} \rightarrow {\mathbf{R}}^{m} \) is convex with respect to componentwise inequality (i.e., the generalized inequality induced by \( {\mathbf{R}}_{ + }^{m} \) ) if and only if for all \( x, y \) and \( 0 \leq \theta ... | \[ f\left( {{\theta x} + \left( {1 - \theta }\right) y}\right) \preccurlyeq {\theta f}\left( x\right) + \left( {1 - \theta }\right) f\left( y\right) ,\] | Yes |
Example 3.49 The quadratic matrix function \( g : {\mathbf{R}}^{m \times n} \rightarrow {\mathbf{S}}^{n} \) defined by\n\n\[ g\left( X\right) = {X}^{T}{AX} + {B}^{T}X + {X}^{T}B + C, \]\n\nwhere \( A \in {\mathbf{S}}^{m}, B \in {\mathbf{R}}^{m \times n} \), and \( C \in {\mathbf{S}}^{n} \), is convex when \( A \succcur... | The function \( h : {\mathbf{S}}^{n} \rightarrow \mathbf{R} \) defined by \( h\left( Y\right) = - \log \det \left( {-Y}\right) \) is convex and increasing on \( \operatorname{dom}h = - {\mathbf{S}}_{+ + }^{n} \) .\n\nBy the composition theorem, we conclude that\n\n\[ f\left( X\right) = - \log \det \left( {-\left( {{X}^... | Yes |
Consider the optimization problem\n\n\\[ \n\\text{minimize}\\,{f}_{0}\\left( x\\right) \n\\]\n\n\\[ \n\\text{subject to}\\,{l}_{i} \\leq {x}_{i} \\leq {u}_{i},\\,i = 1,\\ldots, n\\text{,}\n\\]\n\nwhere \\( x \\in {\\mathbf{R}}^{n} \\) is the variable. The constraints are called variable bounds (since they give lower an... | We can express this problem in standard form as\n\n\\[ \n\\text{minimize}\\,{f}_{0}\\left( x\\right) \n\\]\n\n\\[ \n\\text{subject to}\\,{l}_{i} - {x}_{i} \\leq 0,\\,i = 1,\\ldots, n \n\\]\n\n\\[ \n{x}_{i} - {u}_{i} \\leq 0,\\,i = 1,\\ldots, n. \n\\]\n\nThere are \\( {2n} \\) inequality constraint functions:\n\n\\[ \n{... | Yes |
\[ \text{minimize}\parallel {Ax} - b{\parallel }_{2}\text{,} \] | Since the norm is always nonnegative, we can just as well solve the problem \[ \text{minimize}\parallel {Ax} - b{\parallel }_{2}^{2} = {\left( Ax - b\right) }^{T}\left( {{Ax} - b}\right) \text{,} \] in which we minimize the square of the Euclidean norm. The problems (4.5) and (4.6) are clearly equivalent; the optimal p... | Yes |
Consider a problem with strictly convex quadratic objective, with some of the variables unconstrained:\n\n\[ \text{minimize}\;{x}_{1}^{T}{P}_{11}{x}_{1} + 2{x}_{1}^{T}{P}_{12}{x}_{2} + {x}_{2}^{T}{P}_{22}{x}_{2} \]\n\n\[ \text{subject to}\;{f}_{i}\left( {x}_{1}\right) \leq 0,\;i = 1,\ldots, m\text{,} \] | where \( {P}_{11} \) and \( {P}_{22} \) are symmetric. Here we can analytically minimize over \( {x}_{2} \) :\n\n\[ \mathop{\inf }\limits_{{x}_{2}}\left( {{x}_{1}^{T}{P}_{11}{x}_{1} + 2{x}_{1}^{T}{P}_{12}{x}_{2} + {x}_{2}^{T}{P}_{22}{x}_{2}}\right) = {x}_{1}^{T}\left( {{P}_{11} - {P}_{12}{P}_{22}^{-1}{P}_{12}^{T}}\righ... | Yes |
Consider the problem of minimizing the quadratic function\n\n\[ \n{f}_{0}\left( x\right) = \left( {1/2}\right) {x}^{T}{Px} + {q}^{T}x + r, \n\]\n\nwhere \( P \in {\mathbf{S}}_{ + }^{n} \) (which makes \( {f}_{0} \) convex). The necessary and sufficient condition for \( x \) to be a minimizer of \( {f}_{0} \) is\n\n\[ \... | - If \( q \notin \mathcal{R}\left( P\right) \), then there is no solution. In this case \( {f}_{0} \) is unbounded below.\n\n- If \( P \succ 0 \) (which is the condition for \( {f}_{0} \) to be strictly convex), then there is a unique minimizer, \( {x}^{ \star } = - {P}^{-1}q \).\n\n- If \( P \) is singular, but \( q \... | Yes |
Consider a loss risk constraint of the form\n\n\[ \operatorname{prob}\left( {r \leq \alpha }\right) \leq \beta \]\n\nwhere \( \alpha \) is a given unwanted return level (e.g., a large loss) and \( \beta \) is a given maximum probability. | As in the stochastic interpretation of the robust LP given above, we can express this constraint using the cumulative distribution function \( \Phi \) of a unit Gaussian random variable. The inequality (4.39) is equivalent to\n\n\[ {\bar{p}}^{T}x + {\Phi }^{-1}\left( \beta \right) {\begin{Vmatrix}{\sum }^{1/2}x\end{Vma... | Yes |
Suppose \( y = {Ax} + v \), where \( v \in {\mathbf{R}}^{m} \) is a measurement noise, \( y \in {\mathbf{R}}^{m} \) is a vector of measurements, and \( x \in {\mathbf{R}}^{n} \) is a vector to be estimated, given the measurement \( y \). We assume that \( A \) has rank \( n \), and that the measurement noise satisfies ... | It is a famous result that the problem (4.58) has an optimal solution, the least-squares estimator, or pseudo-inverse,\n\n\[ {F}^{ \star } = {A}^{ \dagger } = {\left( {A}^{T}A\right) }^{-1}{A}^{T}. \]\n\nFor any \( F \) with \( {FA} = I \), we have \( F{F}^{T} \succcurlyeq {F}^{ \star }{F}^{\star T} \). The matrix\n\n\... | Yes |
Minimal upper bound on a set of matrices. We consider the (convex) vector optimization problem, with respect to the positive semidefinite cone,\n\n\\[ \n\\text{minimize (w.r.t.}{\\mathbf{S}}_{ + }^{n}\\text{)}X \n\\]\n\n(4.63)\n\n\\[ \n\\text{subject to}\\;X \\succcurlyeq {A}_{i},\\;i = 1,\\ldots, m\\text{,}\n\\]\n\nwh... | To find a Pareto optimal point, we apply scalarization: we choose any \\( W \\in {\\mathbf{S}}_{+ + }^{n} \\) and form the problem\n\n\\[ \n\\text{minimize}\\;\\operatorname{tr}\\left( {WX}\\right) \n\\]\n\n(4.64)\n\n\\[ \n\\text{subject to}X \\succcurlyeq {A}_{i},\\;i = 1,\\ldots, m\\text{,}\n\\]\n\nwhich is an SDP. D... | Yes |
We consider the problem\n\n\[ \n\\text{minimize}\\;\\left( {1/2}\\right) {x}^{T}{Px} + {q}^{T}x + r \n\]\n\n\[ \n\\text{subject to}\\;{Ax} = b\\text{,}\n\]\n\nwhere \( P \\in {\\mathbf{S}}_{ + }^{n} \) . | The KKT conditions for this problem are\n\n\[ \nA{x}^{ \\star } = b,\\;P{x}^{ \\star } + q + {A}^{T}{\\nu }^{ \\star } = 0,\n\]\n\nwhich we can write as\n\n\[ \n\\left\\lbrack \\begin{matrix} P & {A}^{T} \\\\ A & 0 \\end{matrix}\\right\\rbrack \\left\\lbrack \\begin{matrix} {x}^{ \\star } \\\\ {\\nu }^{ \\star } \\end{... | Yes |
We consider the entropy maximization problem\n\n\[ \n\\text{ minimize }\\;{f}_{0}\\left( x\\right) = \\mathop{\\sum }\\limits_{{i = 1}}^{n}{x}_{i}\\log {x}_{i} \n\]\n\n\[ \n\\text{subject to}\\;{Ax} \\preccurlyeq b \n\]\n\n\[ \n{\\mathbf{1}}^{T}x = 1 \n\]\n\nwith domain \( {\\mathbf{R}}_{+ + }^{n} \), and its dual prob... | Suppose we have solved the dual problem. The Lagrangian at \( \\left( {{\\lambda }^{ \\star },{\\nu }^{ \\star }}\\right) \) is\n\n\[ \nL\\left( {x,{\\lambda }^{ \\star },{\\nu }^{ \\star }}\\right) = \\mathop{\\sum }\\limits_{{i = 1}}^{n}{x}_{i}\\log {x}_{i} + {\\lambda }^{\\star T}\\left( {{Ax} - b}\\right) + {\\nu }... | Yes |
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