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Jesús Rodríguez
Jesús Rodríguez
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Consider a function $f(S): \mathcal{S} \to \mathbb{R}$, where $\mathcal{S}$ is the convex cone of all positive semidefinite complex $M\times M$ Hermitian matrices. The function $f(\cdot )$ is convexconcave over $\mathcal{S}$.

I now define another function $g(L): \mathcal{RSM}(M,N) \to \mathbb{R}$ where $\mathcal{RSM}(M,N)$ is the relaxed Stiefel manifold in $\mathbb{C}^M$, i.e., if $L \in \mathcal{RSM}(M,N)$, then $L^\mathrm{H} L \preceq I_N$. The notation $A \preceq B$ means that $A-B$ is negative semi-definite. The function $g(L)$ is defined as $g(L)=f(LL^{\mathrm{H}})$.

My question is concerning what convexity-properties of $g(\cdot )$ that can be $in \; general$ inherited from $f(\cdot)$. For several different $f(\cdot)$ I have observed that the function $g(\cdot )$ is quasi-convex over $\mathcal{RSM}(M,N)$. Is this true in general?

If no general conclusions can be made, I am in particular interested in the function $$f(S) = \det (I+HSH^\mathrm{H}),$$$$f(S) = \log (\det (I+HSH^\mathrm{H})),$$ where $H$ is an $P \times M$ matrix where $P>N$.

Moreover, I am even more interested in the function $$f(S_1,S_2,\ldots,S_K) = \det (I+\sum_{k=1}^K H_kS_kH_k^\mathrm{H})$$$$f(S_1,S_2,\ldots,S_K) = \log (\det (I+\sum_{k=1}^K H_kS_kH_k^\mathrm{H}))$$ in which case, the notation above needs slight revision, i.e., $f(\cdot,\ldots,\cdot): \mathcal{S}^K \to \mathbb{R}$ etc.

Finally, $A^\mathrm{H}$ denotes the conjugate transpose operator.

Consider a function $f(S): \mathcal{S} \to \mathbb{R}$, where $\mathcal{S}$ is the convex cone of all positive semidefinite complex $M\times M$ Hermitian matrices. The function $f(\cdot )$ is convex over $\mathcal{S}$.

I now define another function $g(L): \mathcal{RSM}(M,N) \to \mathbb{R}$ where $\mathcal{RSM}(M,N)$ is the relaxed Stiefel manifold in $\mathbb{C}^M$, i.e., if $L \in \mathcal{RSM}(M,N)$, then $L^\mathrm{H} L \preceq I_N$. The notation $A \preceq B$ means that $A-B$ is negative semi-definite. The function $g(L)$ is defined as $g(L)=f(LL^{\mathrm{H}})$.

My question is concerning what convexity-properties of $g(\cdot )$ that can be $in \; general$ inherited from $f(\cdot)$. For several different $f(\cdot)$ I have observed that the function $g(\cdot )$ is quasi-convex over $\mathcal{RSM}(M,N)$. Is this true in general?

If no general conclusions can be made, I am in particular interested in the function $$f(S) = \det (I+HSH^\mathrm{H}),$$ where $H$ is an $P \times M$ matrix where $P>N$.

Moreover, I am even more interested in the function $$f(S_1,S_2,\ldots,S_K) = \det (I+\sum_{k=1}^K H_kS_kH_k^\mathrm{H})$$ in which case, the notation above needs slight revision, i.e., $f(\cdot,\ldots,\cdot): \mathcal{S}^K \to \mathbb{R}$ etc.

Finally, $A^\mathrm{H}$ denotes the conjugate transpose operator.

Consider a function $f(S): \mathcal{S} \to \mathbb{R}$, where $\mathcal{S}$ is the convex cone of all positive semidefinite complex $M\times M$ Hermitian matrices. The function $f(\cdot )$ is concave over $\mathcal{S}$.

I now define another function $g(L): \mathcal{RSM}(M,N) \to \mathbb{R}$ where $\mathcal{RSM}(M,N)$ is the relaxed Stiefel manifold in $\mathbb{C}^M$, i.e., if $L \in \mathcal{RSM}(M,N)$, then $L^\mathrm{H} L \preceq I_N$. The notation $A \preceq B$ means that $A-B$ is negative semi-definite. The function $g(L)$ is defined as $g(L)=f(LL^{\mathrm{H}})$.

My question is concerning what convexity-properties of $g(\cdot )$ that can be $in \; general$ inherited from $f(\cdot)$. For several different $f(\cdot)$ I have observed that the function $g(\cdot )$ is quasi-convex over $\mathcal{RSM}(M,N)$. Is this true in general?

If no general conclusions can be made, I am in particular interested in the function $$f(S) = \log (\det (I+HSH^\mathrm{H})),$$ where $H$ is an $P \times M$ matrix where $P>N$.

Moreover, I am even more interested in the function $$f(S_1,S_2,\ldots,S_K) = \log (\det (I+\sum_{k=1}^K H_kS_kH_k^\mathrm{H}))$$ in which case, the notation above needs slight revision, i.e., $f(\cdot,\ldots,\cdot): \mathcal{S}^K \to \mathbb{R}$ etc.

Finally, $A^\mathrm{H}$ denotes the conjugate transpose operator.

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Jesús Rodríguez
Jesús Rodríguez
  • 11
  • 2

Inheriting quasiconvexity from convex function after re-parametrisation of space into the Stiefel manifold

Consider a function $f(S): \mathcal{S} \to \mathbb{R}$, where $\mathcal{S}$ is the convex cone of all positive semidefinite complex $M\times M$ Hermitian matrices. The function $f(\cdot )$ is convex over $\mathcal{S}$.

I now define another function $g(L): \mathcal{RSM}(M,N) \to \mathbb{R}$ where $\mathcal{RSM}(M,N)$ is the relaxed Stiefel manifold in $\mathbb{C}^M$, i.e., if $L \in \mathcal{RSM}(M,N)$, then $L^\mathrm{H} L \preceq I_N$. The notation $A \preceq B$ means that $A-B$ is negative semi-definite. The function $g(L)$ is defined as $g(L)=f(LL^{\mathrm{H}})$.

My question is concerning what convexity-properties of $g(\cdot )$ that can be $in \; general$ inherited from $f(\cdot)$. For several different $f(\cdot)$ I have observed that the function $g(\cdot )$ is quasi-convex over $\mathcal{RSM}(M,N)$. Is this true in general?

If no general conclusions can be made, I am in particular interested in the function $$f(S) = \det (I+HSH^\mathrm{H}),$$ where $H$ is an $P \times M$ matrix where $P>N$.

Moreover, I am even more interested in the function $$f(S_1,S_2,\ldots,S_K) = \det (I+\sum_{k=1}^K H_kS_kH_k^\mathrm{H})$$ in which case, the notation above needs slight revision, i.e., $f(\cdot,\ldots,\cdot): \mathcal{S}^K \to \mathbb{R}$ etc.

Finally, $A^\mathrm{H}$ denotes the conjugate transpose operator.