text:
\section*{Abstract}

A common task in physics, information theory, and other fields is the analysis of properties of subsystems of a given system. Given the covariance matrix $M$ of a system of $n$ coupled variables, the covariance matrices of the subsystems are principal submatrices of $M$. The rapid growth with $n$ of the set of principal submatrices makes it impractical to exhaustively study each submatrix for even modestly-sized systems. It is therefore of great interest to derive methods for approximating the distributions of important submatrix properties for a given matrix.

Motivated by the importance of differential entropy as a systemic measure of disorder, we study the distribution of log-determinants of principal $k\times k$ submatrices when the covariance matrix has bounded condition number. We derive upper bounds for the right tail and the variance of the distribution of minors, and we use these in turn to derive upper bounds on the standard error of the sample mean of subsystem entropy. Our results demonstrate that, despite the rapid growth of the set of subsystems with $n$, the number of samples that are needed to bound the sampling error is asymptotically independent of $n$. Instead, it is sufficient to increase the number of samples in linear proportion to $k$ to achieve a desired sampling accuracy.

\section{Introduction}\label{sec:introduction}

In many fields of study, researchers use matrices to represent systems of interest. Statisticians and data scientists represent large tabular data sets as matrices \cite{Skiena2017}. In network science, it is common to use adjacency matrices to represent the structure of a network \cite{Newman2018}. In dynamical systems, researchers use Jacobian matrices in the study of the linearized dynamics of a system of coupled variables \cite{Strogatz2015}. For networks, dynamical systems, statistical analysis of large data sets, and other applications, it can be insightful (and even necessary) to examine their components (as subnetworks, subsystems, reduced data sets, and so on). Several researchers have used subsystem properties to characterize robustness and other salient properties of dynamical systems \cite{Tononi1994,Tononi1999,Lucia2005,Randles2011,Li2012,Li2016}. Network scientists count and analyze motifs and other subgraphs in networks to characterize a network's structure \cite{Shen-Orr2002,Milo2002}. Several prominent tools in data science are based on linear sketching, an approach to data dimensionality reduction whereby one obtains a reduced data set via matrix multiplication \cite{Woodruff2014,Francis2018} or as a linear combination of submatrices \cite{Sarlos2006,Clarkson2009,Kyrillidis2014}. An example of such a tool for dimensionality reduction is principal component analysis \cite{Liberty2013}.

The various applications of submatrices motivate the mathematical study of their properties. In this paper, we study the distribution of log-determinants of principal submatrices of a positive definite matrix and show that our results lead to controllable sampling guarantees for computing the mean differential entropy of subsystems for a dynamical system. Researchers have studied the differential entropy of subsystems in areas such as physics \cite{Page1993,Sen1996}, biology \cite{Li2012,Li2016}, neuroscience \cite{Tononi1994,Tononi1999,Lucia2005}, computer science \cite{Randles2011}, and coding theory \cite{Koetter2003}.  For example, Tononi et al.\,(1999) computed a measure of network redundancy from the mean differential entropy of its subsystems of fixed size \cite{Tononi1999}. Teschendorff et al.\,(2014) \cite{Teschendorff2014} used differential entropy to define a measure of network robustness for protein-interaction networks .

For several symmetric multivariate distributions, estimates of differential entropy are affine functions of the log-determinant of a system's covariance matrix. Examples include the multivariate normal distribution \cite{Ahmed1989}, the multivariate $t$ distribution \cite{guerrero1996measure,Nadarajah2005}, and the multivariate Cauchy distribution \cite{Nadarajah2005}. For the $n$-variate normal distribution with covariance matrix $M$, for example, the differential entropy is \cite{Ahmed1989}
\begin{align}
    h(M) = \frac{1}{2} \log(\det M) + \frac{n}{2}\left(1 + \log (2\pi)\right)\,,  \label{eq:diff_ent}
\end{align}
where the base of the logarithm can be any finite positive number\footnote{The base $b$ of the logarithm determines the units of entropy. If one chooses $b=2$, one measures entropy values in bits. If one chooses $b=e$, one measures entropy values in nats.}.
The logarithm of the covariance matrix is thus sufficient to approximate the differential entropy of several multivariate distributions.

The principal submatrices of $M$ are covariance matrices of subsystems that correspond to subsets of coupled variables. One can compute the differential entropy of a subsystem by computing $h$ in \Cref{eq:diff_ent} for a principal submatrix of $M$. A system of $n$ coupled variables possesses $\binom{n}{k}\approx \frac{n^k}{k!}$ subsystems of $k$ variables; each of these subsystems corresponds to one of the $\binom{n}{k}$ principal $k\times k$ submatrices of $M$. The exact computation of the distribution of differential subsystem entropy or its moments thus requires one to compute $O(n^k)$ distinct determinants, an infeasible task for large $n$ and $k$. This task can be computationally prohibitive even for modestly-sized systems. To our knowledge, the largest system for which researchers have exactly computed the differential entropy of subsystems is a synthetic network with $n=12$ variables and subsystems with $k\leq 12$ variables \cite{Tononi1999}.

To address this problem, we study the distribution of log-determinants of principal $k\times k$ submatrices. We refer to these log-determinants as \textit{log-minors} of \textit{size} $k$. As we noted above, these log-determinants are sufficient to determine the subsystem entropy for many important multivariate distributions. Knowledge of the properties of this distribution thus enables the derivation of bounds on the sampling error when estimating subsystem entropy in many applications. We show that, given a bound on the condition number of $M$, the standard error of a sample mean of differential entropy is independent of $n$ and sublinear in $k$, implying that one needs a sublinear number of samples in $k$ to ensure a desired accuracy.

Our paper proceeds as follows. In \Cref{sec:notation}, we introduce some notation that we use throughout this paper. In \Cref{sec:results}, we give several upper bounds on the tail and variance of the distribution of log-minors of a positive-definite matrix with bounded condition number. We present proofs for these bounds in \Cref{sec:proofs} and show numerical examples in \Cref{sec:examples}. In \Cref{sec:sampling}, we apply our theorems to provide probabilistic guarantees on the sample mean and relative error, and we discuss implications for the design of practical schemes for estimating mean subsystem entropy. We conclude and discuss possible extensions in \Cref{sec:conclusions}.

\section{Notation}\label{sec:notation}

Let $M \in \R^{n\times n}$ be a positive-definite matrix. Let $\lambda_1(M)\geq\lambda_2(M)\cdots\geq\lambda_n(M)\geq 0$ be the eigenvalues of $M$. Because $M$ is positive definite, it is also nonsingular; its condition number is $\kappa(M) = \lambda_1(M)/\lambda_n(M)$. For a given index set $I\in[n]^k := \{1, \ldots, n\}^k$, the matrix $M_{I} := [M_{i,j}]_{i,j\in I}$ is the corresponding principal submatrix of $M$. For any fixed $k\leq n$, let $\mathcal{A}_k(M)$ denote the set of all such $k\times k$ submatrices of $M$, and let $A_k(M)$ denote a uniformly-random element of this set. We define a random variable $\Y{M}{k} := \log(\det A_k(M))$ and denote its empirical distribution by $\p_{M,k}$. For convenience, we define $\wedge_{n,k}:=\min\{k,n-k\}$.

\section{Bounds on the distribution of log-minors}\label{sec:results}

In this section, we state bounds on the distribution $\p_{M,k}$ for a positive-definite matrix $M$ with bounded condition number. We give upper bounds for the distribution's support, variance, and right tail. We also show that we can improve these bounds if $M$ is diagonal.

\begin{theorem}[Tail and variance bound for log-minors of a positive-definite matrix]
Let $M$ be a positive-definite $n\times n$ matrix with condition number $\kappa(M)\leq\K$. For every $r\geq 0$, we have
\begin{equation}
    \prob\left(|\Y{M}{k} -\mathbbm E [\Y{M}{k}]|\geq r\right)\leq 3\exp \left(-\frac{r}{\log\K}\sqrt{\frac{n}{k(n-k)}}\right)\,.\label{eq:ldi}
\end{equation}
Furthermore, the variance of $\Y{M}{k}$ satisfies
\begin{equation}
    \var(\Y{M}{k})\leq 6\left(\frac{k(n-k)}{n}\right)(\log\K)^2\,.
    \label{eq:ldi_varbound}
\end{equation}
\label{th:ldi}\end{theorem}

\begin{remark}
The tail bound in \Cref{eq:ldi} does not guarantee that $\Y{M}{k}$ concentrates\footnote{Ledoux defined concentration of measure in Ref.\,\cite{Ledoux2001} (on page 3) as follows. Let $(X,d)$ be a metric space with probability measure $\mu$ on Borel sets of $(X,d)$. The \textit{concentration function} is defined as $\alpha_{(X,d,\mu)}:=\sup\{1-\mu(A_r);A\subset X, \mu(A)\geq\frac{1}{2}\}$, where $r>0$ and $A_r:=\{x\in X;d(x,A)<r\}$ is the open $r$-neighborhood of $A$. The measure $\mu$ has \textit{normal concentration} on $(X,d)$ if there are constants $c$ and $C$ such that $\alpha_{(X,d,\mu)}\leq C\exp(-cr^2)$ for every $r$.} on $\mathcal A_k(M)$. This is because the bound in \Cref{eq:ldi} is increasing with respect to $k$ and asymptotically constant with respect to $n$. Indeed, for the bound to approach $0$ for a sequence $\{M_i\}$ of matrices, it is both necessary and sufficient that $\lim_{i\rightarrow \infty} \sqrt{k_i}\log \kappa_i(M) \rightarrow 0$. Because $k$ cannot be smaller than $1$, this condition requires the condition number $\kappa_i(M)$ to approach $1$. The condition $\lim_{i\rightarrow\infty}\kappa_i(M)\rightarrow 1$ severely constrains the sequence $\{M_i\}$. In that limit, all eigenvalues of $M$ are equal to each other and all log-minors are equal to $k\log\lambda_1$.
\end{remark}

\begin{theorem}[Support and variance bound for log-minors of a positive-definite matrix]
Let $M$ be a positive-definite $n\times n$ matrix with condition number $\kappa(M)\leq\K$. For any $k$, the random variable $\Y{M}{k}$ and its distribution $\p_{M,k}$ satisfy the following properties:
\begin{enumerate}
    \item The distribution $\p_{M,k}$ has bounded support that is contained in an interval whose length is no greater than $(\wedge_{n,k}\times\log\K)$; and
    \item the variance of $\Y{M}{k}$ satisfies
    \begin{equation}
        \var(\Y{M}{k})\leq \frac{1}{4}\left(\wedge_{n,k}\times \log \K\right)^2\,.
        \label{eq:support_varbound}
    \end{equation}
\end{enumerate}
\label{th:support}\end{theorem}

\begin{remark}
The variance bound in \Cref{eq:support_varbound} is much sharper than the one in \Cref{eq:ldi_varbound}. Both variance bounds are asymptotically constant with respect to $n$. For fixed $k$, the two variance bounds differ by a factor of $24$ in the large-$n$ limit.
\end{remark}

\begin{remark}
For even $n$ and $k\in\{1,n-1\}$, the bound on the variance in \eq{eq:support_varbound} is sharp when $M$ is a $2\ell\times 2\ell$ (where $\ell=n/2$) diagonal matrix with entries $\lambda_1 = \cdots = \lambda_\ell = \K$ and $\lambda_{\ell+1} = \cdots = \lambda_{2\ell}=1$.
\end{remark}

When $M$ is diagonal, we can derive a variance bound that is sharper than the bounds in \Cref{eq:ldi_varbound,eq:support_varbound}.

\begin{theorem}[Variance bound for log-minors of a positive-definite diagonal matrix]
Let $D$ be a positive-definite $n\times n$ diagonal matrix with condition number $\kappa(D)\leq\K$. The variance of $\Y{D}{k}$ satisfies
\begin{equation}
    \var(\Y{D}{k})\leq\frac{k}{4}\left(\frac{n-k}{n-1}\right)(\log\K)^2\,.
    \label{eq:diag_varbound}
\end{equation}
\label{th:diag}\end{theorem}

\begin{remark}
The two variance bounds in \Cref{eq:support_varbound,eq:diag_varbound} are asymptotically constant with respect to $n$ and converge to the same limiting value of $k(\log\K)^2/4$.
\end{remark}

\begin{remark}
The variance bound for diagonal positive-definite matrices in \Cref{eq:diag_varbound} is sharper than the variance bound for general positive-definite matrices in \Cref{eq:support_varbound}. The former differs from the latter by a factor of $\max\{k,n-k\}/(n-1)\leq 1$.
\end{remark}

\begin{remark}
For even $n$ and any $k\leq n$, the bound on the variance in \eq{eq:diag_varbound} is sharp when $M$ is a $2\ell\times 2\ell$ (where $\ell=n/2$) diagonal matrix with entries $\lambda_1 = \cdots = \lambda_\ell = \K$ and $\lambda_{\ell+1} = \cdots = \lambda_{2\ell}=1$. The sharpness of the bound for diagonal matrices indicates a limit to possible improvements for the variance bound for general positive-definite matrices. Specifically, one cannot hope to improve the variance bound in \Cref{eq:support_varbound} by more than a factor of $\max\{k,n-k\}/(n-1)$.
\end{remark}

The variance bound in \Cref{th:support} is sharp for a diagonal matrix. This observation and several examples in \Cref{sec:examples} motivate the following conjecture.

\begin{conj}[Diagonal matrices maximize log-minor variance] Let $\mathcal M_{\kappa}$ be the set of positive-definite $n\times n$ matrices with condition number $\kappa$. For all $k<n$, $\kappa$, and $M\in\mathcal M_{\kappa}$, there exists a diagonal matrix $D\in\mathcal M_{\kappa}$ such that
\begin{equation}
    \var(\Y{M}{k})
        \leq
    \var(\Y{D}{k})\,.
\label{eq:ineq}
\end{equation}
    \label{conj:diag}
\end{conj}

The variance bounds (see \,\Cref{eq:support_varbound,eq:ldi_varbound,eq:diag_varbound}) have important implications for the accuracy of sample means of log-minors. We discuss these implications in \Cref{sec:sampling}.

\section{Proofs of bounds on the distribution of log-minors}\label{sec:proofs}

\subsection{Proof of \Cref{th:ldi}}\label{sec:proof2}

To prove \Cref{th:ldi}, we use Cauchy's interlacing theorem and results on Markov chains on countable sets. Chatterjee and Ledoux (2009) previously used this approach to prove a concentration result for empirical cumulative eigenvalue spectra of Hermitian matrices \cite{Chatterjee2009}.

\begin{prop}[Cauchy's interlacing theorem \cite{Hwang2004}]\label{prop:cauchy}
Let $M$ be a Hermitian $n\times n$ matrix, and let $A$ be a principal $(n-1)\times (n-1)$ submatrix of $M$.
If $M$ has eigenvalues $\lambda_1(M)\geq\lambda_2(M)\geq\dots\geq\lambda_n(M)$ and $A$ has eigenvalues $\lambda_1(A)\geq\lambda_2(A)\geq\dots\geq\lambda_{n-1}(A)$, then
\begin{equation}
    \lambda_1(M)\geq\lambda_1(A)\geq\lambda_2(M)\geq\lambda_2(A)\geq\lambda_3(M)\geq\dots\geq\lambda_{n-1}(M)\geq\lambda_{n-1}(A)\geq\lambda_n(M)\,.\nonumber
\end{equation}
\end{prop}

\begin{prop}[Large-deviation inequality for functions on countable sets \cite{Ledoux2001} (page 50)]\label{prop:ledoux}
Let $(\Pi,\mu_{\mathcal A})$ be a reversible Markov chain on a finite or countable set $\mathcal A$. Let $(\Pi,\mu_{\mathcal A})$ have a spectral gap\footnote{The spectral gap (also called the ``Poincar\'e constant'') of a Markov chain $(\Pi,\mu_{\mathcal A})$ on a space $\mathcal A$ is the constant $\overline{\lambda}$ such that, for all functions $f$, we have $\overline{\lambda}\times\var_\mu(f)\leq\frac{1}{2}\sum_{A,B\in\mathcal A}[f(A)-f(B)]^2\Pi(A,B)\mu(\{A\})$\,. See, for example, Ref.\,\cite{Ledoux2001} (page 50).} of $\overline{\lambda}>0$. It follows that , whenever $f: \mathcal A\rightarrow\mathbbm R$ is a function such that
\begin{equation}
    |||f|||^2_\infty:=\frac{1}{2}\sup_{A\in \mathcal A}\sum_{B\in \mathcal A} \left(f(A)-f(B)\right)^2\Pi(A,B)\leq 1\,,\nonumber
\end{equation}
 it is also true that $f$ is integrable with respect to $\mu_{\mathcal A}$ and that, for every $r\geq 0$, the probability measure
\begin{equation}
    \mu\left(\{f\geq \int_{\mathcal A} f\,d\mu_{\mathcal A} + r\}\right)\leq 3e^{-\frac{r}{2}\sqrt{\overline{\lambda}}}\,. \nonumber
\end{equation}
\end{prop}

\begin{remark}
The expected squared distance in $f$ between $A\in\mathcal A$ and its adjacent states in the Markov chain is $\sum_{B\in\mathcal A}\left(f(A)-f(B)\right)^2\Pi(A,B)$. One can thus think of $|||f|||_{\infty}^2$ as a measure of the expected squared distance between the greatest ``outlier'' $A$ and adjacent states in $\mathcal A$. We thus refer to $|||f|||_{\infty}^2$ as the \textit{squared outlier deviation} of $f$ on $(\Pi,\mu_{\mathcal A})$.
\end{remark}

\begin{prop}[Spectral gap of random-transposition walk \cite{Diaconis1981}]\label{prop:gap}
Let $\mathcal S_n$ be the set of permutations of $n$ elements, and let $s\in\mathcal S_n$. Let the ``random-transposition walk'' be a reversible Markov chain $(\Pi, \mu_{\mathcal S_n})$ with kernel
\begin{align}\Pi=\left\{\begin{matrix}
    1/n\,,&\textrm{if } s'=s\,,\\
2/n^2\,,&\textrm{if } s'=\tau s \textrm{ for some transposition }\tau\,,\\
0\,,&\textrm{otherwise}\,.
    \end{matrix}\right. \nonumber
\end{align}
The random-transposition walk has a spectral gap of $\overline \lambda=2/n$.
\end{prop}

\begin{proof}[Proof of \Cref{th:ldi}]
Every principal $k\times k$ submatrix of $M$ is the top-left principal $k\times k$ submatrix of $M$ after a permutation of its rows and columns. We denote the permutated matrix by $s M$ and its top-left principal $k\times k$ submatrix by $\hat A_k(s M)$, where $s\in \mathcal S_n$ is a permutation of $n$ elements.

For the top-left principal $k\times k$ submatrix, only the first $k$ elements of $s$ are relevant. There are $(n-k)!$ permutations $s\in \mathcal S_n$ that are identical in their first $k$ elements, so there is a $1$-to-$(n-k)!$ correspondence between $\mathcal A_k(M)$ and $\mathcal S_n$. Because of the correspondence between $\mathcal A_k(M)$ and $\mathcal S_n$, we obtain the same distribution for a function $f(A_k(M))$, where we choose $A_k(M)$ uniformly at random from $\mathcal A_k(M)$, and for $f(\hat A_k(sM))$, where we choose $s$ uniformly at random from $\mathcal S_n$.

Let $f_{\alpha}: \mathcal S_n \rightarrow \mathbbm R$ be such that
\begin{equation} \label{eq:fdef}
    f_{\alpha}(s) := \alpha\ld{\hat A_k(sM)}
\end{equation}
for some $\alpha\in\mathbbm R$. To find an upper bound on the squared outlier deviation for $f_{\alpha}$ on the random-transposition walk, we make two observations:
\begin{enumerate}
\item Consider two permutations, $s$ and $s'$, that are adjacent in the random-transposition walk; that is, $s'=\tau s$ for some transposition $\tau$. The determinant is invariant under basis transformation, so the value of $f_{\alpha}(s')$ can differ from $f_{\alpha}(s)$ only if $\tau$ is a transposition that swaps one of the first $k$ elements in $s$ with one of the last $n-k$ elements in $s$. There are $n^2$ possible transpositions for a sequence of $n$ elements; and $2k(n-k)$ of these transpositions swap one of the first $k$ elements of the sequence with one of the last $n-k$ elements of the sequence. Consequently, the fraction of transpositions that change the value of $f_{\alpha}$ has an upper bound of
\begin{equation}
    b_1:= \frac{2k}{n^2}(n-k)\,.\nonumber
\end{equation}
\item Using Cauchy's interlacing theorem (see \Cref{prop:cauchy}), one can find an upper bound $b_2$ for $|f_{\alpha}(A)-f_{\alpha}(B)|$. For any $k<n$ and any pair $A, B\in\mathcal A_k(M)$, there exists a matrix $C\in A_{k+1}(M)$ such that $A$ and $B$ are principal submatrices of $C$. Cauchy's interlacing theorem implies that
\begin{enumerate}
    \item $\lambda_1(M)$ is an upper bound on the largest eigenvalue of $C$;
    \item $\lambda_n(M)$ is a lower bound on the smallest eigenvalue of $C$;
    \item $\sum_{i=1}^{k}\log\lambda_i(C)$ is an upper bound on $\ld{A}$ and $\ld B$; and
    \item $\sum_{i=2}^{k+1}\log\lambda_i(C)$ is a lower bound on $\ld{A}$ and $\ld B$.
\end{enumerate}
Therefore,
\begin{align*}
    |\ld{A}-\ld B|\leq & \sum_{i=1}^{k}\log\lambda_i(C) - \sum_{i=2}^{k+1}\log\lambda_i(C)\\
    \leq & \log\lambda_1(M) - \log\lambda_n(M) \leq \log \K\,.
\end{align*}
This upper bound for $|\ld{A}-\ld B|$ holds for arbitrary $A,B\in\mathcal A_k(M)$. We can thus set the upper bound to be $b_2=\alpha\log\K$.
\end{enumerate}
We obtain an upper bound for the squared outlier deviation of $f_{\alpha}$ of
\begin{align*}
    |||f_{\alpha}|||_\infty^2 &\leq \frac{1}{2}b_1b_2^2\\
&=k(n-k)\left(\frac{\alpha\log\K}{n}\right)^2\,.
\end{align*}

Let $\alpha'=n\log\K/\sqrt{k(n-k)}$. The function $f_{\alpha'}$ on $(\Pi,\mu_{\mathcal S_n})$ has a squared outlier deviation of $|||f_{\alpha'}|||^2_\infty\leq 1$. We can thus use the tail bound for functions on countable sets (see \Cref{prop:ledoux}) for $f_{\alpha'}$. Therefore,
\begin{equation*}
    \prob(|f_{\alpha'}(\hat A_k(\sigma M)) -\mathbbm E[f_{\alpha'}(A_k(\sigma M)]|\geq \alpha'r)\leq 3e^{-\frac{\alpha'r}{2}\sqrt{\overline{\lambda}}}\,.
\end{equation*}
We can substitute $f_{\alpha'}(\hat A_k(\sigma M))$ in \eq{eq:fdef} by $\alpha'\Y{M}{k}$, because of the correspondence between $\mathcal A_k(M)$ and $\mathcal S_n$. Applying \Cref{prop:gap}, we obtain
\begin{align*}
    \prob(|\alpha'\Y{M}{k} -\alpha'\mathbbm E [\Y{M}{k}]\geq \alpha'r)&\leq 3e^{-\frac{\alpha'r}{\sqrt{2n}}}\\
    \implies\prob(|\Y{M}{k} -\mathbbm E[\Y{M}{k}]|\geq r)&\leq 3\exp \left(-\frac{r}{\log\K}\sqrt{\frac{n}{k(n-k)}}\right)\,.
\end{align*}
This proves the first statement of \Cref{th:ldi} (see \eq{eq:ldi}).

We derive a bound on the variance of $\ld{A}$ from \Cref{eq:ldi} from a direct calculation. First, we write
\begin{align*}
    \var(\Y{M}{k})  &= \mathbbm{E}[(\Y{M}{k}-\mathbbm{E}[\Y{M}{k}])^2]\\
    &= \int_{0}^\infty \prob(\Y{M}{k}-\mathbbm{E}[\Y{M}{k}])^2 \geq u)du\\
    &= \int_{0}^\infty \prob(\Y{M}{k}-\mathbbm{E}[\Y{M}{k}])\geq \sqrt u\} du\,.
\end{align*}
Using the tail bound in \eq{eq:ldi}, it follows that
\begin{align*}
    \var(\Y{M}{k})  &\leq \int_{0}^\infty3\exp \left(-\frac{\sqrt{u}}{\log\K}\sqrt{\frac{n}{k(n-k)}}\right) du\\
    & = 6\left(\frac{k(n-k)}{n}\right)(\log\K)^2
\end{align*}
\end{proof}

\subsection{Proof of \Cref{th:support}}\label{sec:proof1}

We prove \Cref{th:support} using Cauchy's interlacing theorem and Popoviciu's inequality.
\begin{prop}[Popoviciu's inequality \cite{Popoviciu1935, Sharma2010}]\label{prop:popoviciu}
Let $X$ be a real-valued random variable supported on the interval $[x_\mathrm{min}, x_{\mathrm{max}}]$. It then follows that $X$ has variance
\begin{equation}
    \var(X)\leq (x_{\operatorname{max}}-x_{\operatorname{min}})^2/4\,.\nonumber
\end{equation}
\end{prop}
For a proof of this version of Popoviciu's inequality, see Ref.\,\cite{stackPopoviciu}.

\begin{proof}[Proof of \Cref{th:support}]
For any finite $n$ and $k$, the set $\mathcal A_k(M)$ of principal $k\times k$ submatrices of an $n\times n$ matrix $M$ has finite cardinality $\binom{n}{k}$. It follows that the distribution of any function of $A_k(M)$ has finite support. We define an interval $[r_1,r_2]$ with $r_1:=\min\Y{M}{k}$ and $r_2:=\max\Y{M}{k}$, such that the support of $\p_{M,k}$ is a finite subset of $[r_1,r_2]$.

We can obtain any principal $k\times k$ submatrix $A$ of $M$ by removing $n-k$ row--column pairs from $M$. Successive applications of Cauchy's interlacing theorem show that $\lambda_1(A)\leq\lambda_1(M)$ and $\lambda_k(A)\geq\lambda_n(M)$. It follows that
\begin{align}
    r_1=k\log\lambda_n(M)\,, \qquad
    r_2=k\log\lambda_1(M)\,.\nonumber
\end{align}
Therefore,
\begin{align}
    r_2-r_1\leq k\left(\log\K+\log\lambda_n(M)\right)-k\log\lambda_n(M)=k\log \K\,.\nonumber
\end{align}
If $k>n/2$, any two principal $k\times k$ submatrices share $2k-n$ rows and columns. They can thus differ in at most $n-k$ rows and columns. It follows that one can refine the lower and upper bounds on the support of $\p_{M,k}$ so that $[r_1,r_2]\leq\wedge_{n,k}\times\log\K$. We have thus proven the first part of Theorem 1. Applying Popiviciu's inequality to $X=\Y{M}{k}$ with $x_{\operatorname{max}}-x_{\operatorname{min}}\leq r_2-r_1$ yields the variance bound in \Cref{th:support}.
\end{proof}

\subsection{Proof of \Cref{th:diag}}\label{sec:proof3}
For our proof of \Cref{th:diag}, we maximize $\var(\Y{D}{k})$ with respect to the eigenvalues of $D$.

\begin{proof}[Proof of \Cref{th:diag}]
Let $D$ be a positive-definite diagonal matrix with entries $\lambda_1(D)\geq\lambda_2(D)\geq\dots\geq\lambda_n(D)>0$.
Define $x_i = \log \lambda_i(D)$ for each $i \in [n]$; and let $I \in [n]^k$. It then follows that
\begin{align}
    \ld{D_{I}} &= \sum_{i \in I}x_{i}\,.\label{eq:xi}
\end{align}
We now consider the function $v(x_1,x_2,\ldots,x_n) := \var(\Y{D}{k})$. From \Cref{eq:xi}, we see that every value of $\Y{D}{k}$ is a sum of a subset of the variables $x_i$. Therefore, the function $v(x_1,x_2,\ldots,x_n)$ is convex (i.e., concave up) in the variables $x_1,x_2,\ldots,x_n$. Furthermore, the variance is translation-invariant. We may therefore, without loss of generality, suppose that $x_n = 0$ (corresponding to $\lambda_n(D) = 1$) and $x_1 = \log\kappa(D)$ (corresponding to $\lambda_n(D) =\kappa(D)$). Consequently, the maximization of the variance $\var(\Y{D}{k})$ amounts to the maximization of $v$ over the volume $[0,\log \kappa(D)]^n$ associated with an $n$-dimensional hypercube with edge length $\log \kappa(D)$. The solutions lie at the vertices of this hypercube. Therefore,
\begin{align*}
    x_i = \begin{cases}
        \log \kappa(D)\,, &\quad i \leq \ell \,, \\
        0 \,, &\quad \text{otherwise}\,,
    \end{cases}
\end{align*}
for some $\ell \in [n]$. We may now view $\Y{D}{k}/(\log \kappa(D))$ as a hypergeometric random variable on a population of size $n$ for which $\ell$ elements have the value $1$ and $n-l$ elements have the value $0$. The variance of this hypergeometric random variable is
\begin{align*}
    \var(\Y{D}{k}) = \frac{k(n-k)}{n^2(n-1)}\left(n-\ell\right)\ell (\log \kappa(D))^2\,,
\end{align*}
which is maximal at
\begin{align*}
    \ell^* =
    \begin{cases}
        \frac{n}{2} \,, &\quad n \text{ even}\,, \\
        \frac{n\pm1}{2}\,, &\quad n\text{ odd}\,.
    \end{cases}
\end{align*}
The maximal value of $\var(\Y{D}{k})$ for even $n$ leads to the variance bound
\begin{align} \label{eq:vbd}
    \var (\Y{D}{k}) &\leq \frac{k(n-k)}{4(n-1)}(\log \kappa(D))^2 \\
    &\leq\frac{k(n-k)}{4(n-1)}(\log \K)^2\,.
\end{align}
Comparing the maximal values of $\var(\Y{D}{k})$ for even $n$ and for odd $n$ shows that \Cref{eq:vbd} is a variance bound for all $n$.
\end{proof}

\section{Examples}\label{sec:examples}

In this section, we compare the tail of the distribution $\p_{M,k}$ for several example matrices $M$ to the bounds in Theorems \ref{th:ldi} and \ref{th:diag}.

We consider four examples of positive-definite $n\times n$ matrices with $n=20$ and fixed condition number $\kappa=3$. \paragraph{Example E1.} Consider the diagonal matrix $M_{\textrm{E1}}$ that maximizes the variance of $\Y{M_{\textrm{E1}}}{k}$.  (See the proof of \Cref{th:diag}.) For even $n$, this matrix has eigenvalues $\lambda_1, \lambda_2, \ldots, \lambda_n$, where $\lambda_1,\ldots,\lambda_{n/2}=\K$ and $\lambda_{n/2+1},\ldots,\lambda_n=1$.

\paragraph{Example E2.} Consider a diagonal matrix $M_{\textrm{E2}}$ with eigenvalues $\lambda_1, \lambda_2, \ldots, \lambda_n$. We set $\lambda_1=\kappa$ and $\lambda_n=1$. We draw $\lambda_2,\ldots,\lambda_{n-1}$ from a uniform distribution on $[1,\kappa]$.

\paragraph{Example E3.} We obtain a non-diagonal positive-definite matrix $M_{\textrm{E3}}$ with condition number $\kappa$ via an orthogonal transformation of $M_{\textrm{E1}}$. That is,
\begin{equation}
    M_{\textrm{E3}} := Q^{-1}M_{\textrm{E1}}Q\,,\nonumber
\end{equation}
where $Q$ is an orthogonal matrix that we choose from the Haar measure over the group of orthogonal matrices. We use Stewart's algorithm \cite{Stewart1980} to generate $Q$.

\paragraph{Example E4.} We again generate a random orthogonal matrix $Q$ using Stewart's algorithm. We obtain another non-diagonal positive-definite matrix $M_{\textrm{E4}}:=Q^{-1}M_{\textrm{E2}}Q$ via an orthogonal transformation of $M_{\textrm{E2}}$.\vspace{1em}

\begin{figure}[p!tb]
\centering
\includegraphics[trim={0cm 1.5cm 0cm 2.75cm},clip,width=1\textwidth]{fig1}
\caption{Approximate probability density functions for log-minors. We show $\p_{M,k}$ for Examples E1, E2, E3, and E4 with $n=20$ and condition number $\kappa=3$. We show results for $k=1$ in panels \panel{A} and \panel{B}, for $k=5$ in panels \panel{C} and \panel{D}, for $k=10$ in panels \panel{E} and \panel{F}, and for $k=19$ in panels \panel{G} and \panel{H}.}
\label{fig:pdfs}
\end{figure}

In \Cref{fig:pdfs}, we show the empirical probability densities of $\Y{M}{k}$ for Examples E1, E2, E3, and E4 using four different values of $k$. For all four examples, we observe that the interval on which $\p_{M,k}$ is supported shifts to the right for progressively larger $k$. The length of the supported interval increases with $\wedge_{n,k}$. For $k=5$ and $k=10$ --- the cases in which $\wedge_{n,k}$ is larger than $1$ --- the distribution $\p_{M,k}$ are almost symmetric about $\E[\Y{M}{k}]$ for all four examples. For Example E1, the distribution $\p_{M,k}$ is symmetric about its mean for all examined values of $k$. Its density is nonzero at $\wedge_{n,k}+1$ equidistant points.

\begin{table}[h!]
\begin{center}
\begin{tabular}{ccccccc}
    \hline\rule{0pt}{2.6ex}\rule[-1.2ex]{0pt}{0pt}
    \multirow{2}{*}{$k$}
    & \multirow{2}{*}{Example}
    & \multirow{2}{*}{$\mathbbm E[\Y{M}{k}]$}
    & \multirow{2}{*}{$\var(\Y{M}{k})$}
    & \multicolumn{3}{c}{Variance bound from} \\
    & & &
    & Theorem 1 & Theorem 2 & Theorem 3 \\
    \hline\rule{0pt}{2.6ex}\rule[-1.2ex]{0pt}{0pt}
    1 & E1 & $0.549$ & $0.302$ & $6.880$ & $0.302$ & $0.302$ \\\rule{0pt}{2.6ex}\rule[-1.2ex]{0pt}{0pt}
     & E2 & $0.689$ & $0.115$ & $6.880$ & $0.302$ & $0.302$ \\\rule{0pt}{2.6ex}\rule[-1.2ex]{0pt}{0pt}
     & E3 & $0.683$ & $0.021$ & $6.880$ & $0.302$ & N/A \\\rule{0pt}{2.6ex}\rule[-1.2ex]{0pt}{0pt}
     & E4 & $0.739$ & $0.005$ & $6.880$ & $0.302$ & N/A\\\rule{0pt}{2.6ex}\rule[-1.2ex]{0pt}{0pt}
    5 & E1 & $2.747$ & $1.191$ & $27.156$ & $1.509$ & $1.191$ \\\rule{0pt}{2.6ex}\rule[-1.2ex]{0pt}{0pt}
     & E2 & $3.446$ & $0.454$ & $27.156$ & $1.509$ & $1.191$ \\\rule{0pt}{2.6ex}\rule[-1.2ex]{0pt}{0pt}
     & E3 & $3.283$ & $0.091$ & $27.156$ & $1.509$ & N/A \\\rule{0pt}{2.6ex}\rule[-1.2ex]{0pt}{0pt}
     & E4 & $3.649$ & $0.020$ & $27.156$ & $1.509$ & N/A \\\rule{0pt}{2.6ex}\rule[-1.2ex]{0pt}{0pt}
    10 & E1 & $5.493$ & $1.588$ & $36.208$ & $3.017$ & $1.588$ \\\rule{0pt}{2.6ex}\rule[-1.2ex]{0pt}{0pt}
     & E2 & $6.893$ & $0.605$ & $36.208$ & $3.017$ & $1.588$ \\\rule{0pt}{2.6ex}\rule[-1.2ex]{0pt}{0pt}
     & E3 & $6.213$ & $0.128$ & $36.208$ & $3.017$ & N/A \\\rule{0pt}{2.6ex}\rule[-1.2ex]{0pt}{0pt}
     & E4 & $7.176$ & $0.031$ & $36.208$ & $3.017$ & N/A \\\rule{0pt}{2.6ex}\rule[-1.2ex]{0pt}{0pt}
    19 & E1 & $10.437$ & $0.302$ & $6.880$ & $0.302$ & $0.302$ \\\rule{0pt}{2.6ex}\rule[-1.2ex]{0pt}{0pt}
     & E2 & $13.096$ & $0.115$ & $6.880$ & $0.302$ & $0.302$ \\\rule{0pt}{2.6ex}\rule[-1.2ex]{0pt}{0pt}
     & E3 & $10.570$ & $0.022$ & $6.880$ & $0.302$ & N/A \\\rule{0pt}{2.6ex}\rule[-1.2ex]{0pt}{0pt}
     & E4 & $13.155$ & $0.008$ & $6.880$ & $0.302$ & N/A \\
\hline
\end{tabular}
\end{center}
\vspace{-0.5cm}
\caption{Expectation and variance for $\Y{M}{k}$ for Examples E1, E2, E3, and E4. For comparison, we show the numerical values of the variance bounds from \Cref{th:ldi} (see \Cref{eq:ldi_varbound}) and \Cref{th:support} (see \Cref{eq:support_varbound}). For the examples with diagonal matrices (E1 and E2), we also show the numerical value of the variance bound from \Cref{th:diag} (see \Cref{eq:diag_varbound}).}
\label{tab:examples}
\end{table}

In \Cref{tab:examples}, we show $\E[\Y{M}{k}]$ and $\var(\Y{M}{k})$ for the distributions in \Cref{fig:pdfs}. We first consider the expectation of $\Y{M}{k}$. For all four examples, we observe that $\E[\Y{M}{k}]$ increases with $k$. For all examined
values of $k$, we see that $\E[\Y{M_{\textrm{E4}}}{k}]>\E[\Y{M_{\textrm{E2}}}{k}]>\E[\Y{M_{\textrm{E3}}}{k}]>\E[\Y{M_{\textrm{E1}}}{k}]$. Our observations thus suggest that the expectation of $\Y{M}{k}$ is large when we choose eigenvalues of $M$ uniformly at random from the interval $[1,\kappa]$ and small when we set half of the eigenvalues of $M$ to $1$ and the other half to $\kappa$.

We now give several observations about the variance of $\Y{M}{k}$. For all examined values of $k$, we see that $\var(\Y{M_{\textrm{E1}}}{k})>\var(\Y{M_{\textrm{E2}}}{k})>\var(\Y{M_{\textrm{E3}}}{k})>\var(\Y{M_{\textrm{E4}}}{k})$. Our observation of larger $\var(\Y{M}{k})$ for the examples with diagonal matrices (Examples E1 and E2) than for the examples with non-diagonal matrices (Examples E3 and E4) gives intuitive support for \Cref{conj:diag}. Our observation that $\var(\Y{M_{\textrm{E1}}}{k})>\var(\Y{M_{\textrm{E2}}}{k})$ reflects the fact that Example E1 maximizes the variance in this case (see \Cref{th:diag}).

For all examined $k$, the value of the variance bound in \Cref{th:ldi} (see \Cref{eq:ldi_varbound}) is at least 12 times larger than the value of the variance bound in \Cref{th:support} (see \Cref{eq:support_varbound}). For $k=1$ and $k=19$, the cases in which $\wedge_{n,k}=1$, the value of the variance bound in \Cref{th:support} is equal to the value of the variance bound for diagonal positive-definite matrices (\Cref{th:diag}). Additionally, it is sharp in Example E1.

\begin{figure}[t]
\centering
\includegraphics[trim={0cm 0.25cm 0cm 0.5cm},clip,width=1\textwidth]{fig2}
\caption{Empirical tails of the distribution of log-minors. We show $\prob(|\Y{M}{k}-\mathbbm E[\Y{M}{k}]|\geq r)$ for positive-definite $n\times n$ matrices from examples E1, E2, E3, and E4 with $n=20$ and condition number $\kappa=3$. We show results for $k\times k$ submatrices with \panel{A} $k=1$, \panel{B} $k=5$, \panel{C} $k=10$, and \panel{D} $k=19$. The curve B1 represents the tail bound from \Cref{th:ldi}. The curves B2 and B3 visualize Chebyshev bounds that we obtain from the variance bounds in \Cref{th:support,th:diag}, respectively.}
\label{fig:large_deviations}
\end{figure}

In Fig.\,\ref{fig:large_deviations}, we show the empirical tails $\prob(|\Y{M}{k}-\mathbbm E[\Y{M}{k}]|\geq r)$ for our four examples. We also show the tail bound B1 from \Cref{th:ldi} and two Chebyshev bounds\footnote{We can obtain a tail bound from a variance bound by using Chebyshev's inequality \cite{Blitzstein2014} (page 429), $\operatorname{Pr}(|X-\mathbbm E[X]|\geq r)\leq \frac{\var(X)}{r^2}$, for an integrable random variable $X$ and $r\in\mathbbm R_+$.}, B2 and B3, which we obtain from the variance bounds in \Cref{th:support,th:diag}, respectively. Consistent with our observations in \Cref{tab:examples} on $\var(\Y{M}{k})$, we observe that the tail probability tends to be larger for the examples with diagonal matrices (Examples E1 and E2) than for the examples with non-diagonal matrices (Examples E3 and E4).

The difference in functional form guarantees that the bound B1 intersects with the Chebyshev bound B2 at two values of $r$. If we denote these values by $r'$ and $r''>r'$, the bound B1 is sharper than B2 on $[0,r']$ and $[r'',\infty)$. In our observations, both bounds exceed the trivial bound $\prob(|\Y{M}{k}-\mathbbm E[\Y{M}{k}]|\geq r)\leq 1$ on $[0,r']$. The value $r''$ lies outside the support of $\prob(|\Y{M}{k}-\mathbbm E[\Y{M}{k}]|\geq r)$. We thus see that B1 is sharper than B2 only for values of $r$ for which neither bound is informative.

For $k=1$ and $k=19$, the bounds B2 and B3 coincide and are sharp at $r=(\log\kappa)/2$ when $M=M_{\textrm{E1}}$. For $k=5$ and $k=10$, the bound (B3) for diagonal positive-definite matrices is sharper than the bound (B2) for general positive-definite matrices. The difference between the two bounds is most visible for $k=10$, which is the case that maximizes $\wedge_{n,k}$.

\section{Estimating mean subsystem entropy}\label{sec:sampling}

We now consider the implications of our results in Section \ref{sec:results} for the problem of estimating the mean subsystem entropy of a given system of coupled variables. When the joint distribution of variables is a multivariate normal distribution, one can compute the differential entropy of a subsystem by applying \Cref{eq:diff_ent} to the corresponding sub-covariance matrix. We are interested in the \emph{mean subsystem entropy} $\E[h(A_k(M))]$ for subsystems of $k$ variables. As we noted previously, the large number of subsystems for even modest values of $n$ and $k$ render it prohibitive to exactly compute $\E[h(A_k(M))]$. Fortunately, the tail and variance bounds in Section \ref{sec:results} allow us to instead provide sampling guarantees, through which one can achieve a prescribed sampling accuracy. We give upper bounds on  the standard error and on the coefficient of variation for both a sample mean of $\Y{M}{k}$ and a sample mean of subsystem entropy.

Fix a subsystem size $k$  and sample size $q\geq 1$. The $q$-sample mean of $\Y{M}{k}$ is
\begin{align*}
    S_Y := \frac{1}{q}\sum_{i = 1}^q \ld{A_i}\,,
\end{align*}
where we choose each $A_i$ uniformly at random from $\mathcal A_k(M)$. The $q$-sample mean of subsystem entropy is
\begin{align*}
    S_h := \frac{1}{q}\sum_{i = 1}^q h(A_i)\,.
\end{align*}
We use $S_Y$ and $S_h$ as estimators of the population means $\E[\Y{M}{k}]$ and $\E[h(A_k(M))]$, respectively. These estimators are unbiased, as $\E[S_Y]=\E[\Y{M}{k}]$ and $\E[S_h]=\E[h(A_k(M))]$. A measure of reliability of an estimator is the \emph{standard error}, which one computes as the estimator's standard deviation. Because $h(A_k(M))$ differs from $\Y{M}{k}/2$ by a constant, the sample mean $S_h$ has the standard error
\begin{align*}
    \hat{\sigma}(S_h) = \frac{1}{2}\hat{\sigma}(S_Y) = \frac{1}{2}\sqrt{\frac{\var \left(\Y{M}{k}\right)}{q}}\,.
\end{align*}
We may therefore use the bounds of \Cref{th:ldi,th:support,th:diag} to derive bounds on the standard error for $S_Y$ and $S_h$.

\begin{cor}[Standard error of the sample mean subsystem entropy]
Let $M$ be a covariance matrix of an $n$-variate normal distribution, and suppose that the condition number of $M$ satisfies $\kappa(M)\leq\K$. Let $S_h$ be the $q$-sample mean of the entropy of subsets of $k$ variables; and let $S_Y$ be the $q$-sample mean of log-determinants of $k\times k$ principal submatrices of $M$. It then follows, for any subsystem size $k$, that the standard error of the mean subsystem entropy is $\hat\sigma(S_h)=\hat\sigma(S_Y)/2$ and that $\hat\sigma(S_Y)$ satisfies
\begin{equation}
    \hat{\sigma}(S_Y)\leq\sqrt{\frac{6k(n-k)}{qn}}\log \K\,,\label{eq:se1}
\end{equation}
and
\begin{equation}
    \hat{\sigma}(S_Y)\leq\frac{1}{2}\sqrt{\frac{\wedge_{n,k}}{q}}\log \K\,.\label{eq:se2}
\end{equation}
Furthermore, if $M$ is diagonal,
\begin{equation}
    \hat{\sigma}(S_Y)\leq\frac{1}{2}\sqrt{\frac{k(n-k)}{q(n-1)}}\log\K\,.\label{eq:se3}
\end{equation}
\end{cor}

The \emph{coefficient of variation} $\cv{S}$ is another measure of reliability for estimators. It measures the size of the typical error of an estimator $S$ as a fraction of the magnitude of $\E[S]$. As a formula, it is given by
\begin{align}
    \cv{S} := \frac{\hat{\sigma}(S)}{\E[S]}\,. \label{eq:cv_def}
\end{align}
The coefficient of variation for $S_Y$ arises from the standard deviation of the relative error
\begin{align*}
    \mathcal{E} := \abs{\frac{\Y{M}{k} - \E[\Y{M}{k}]}{\E[\Y{M}{k}]}}\,
\end{align*}
of $\Y{M}{k}$ because
\begin{align*}
    \frac{\hat{\sigma}(S_Y)}{\E[S_Y]}
    =\frac{\hat{\sigma}(\Y{M}{k})- \E[\Y{M}{k}]}{\E[\Y{M}{k}]\sqrt{q}}
    =\hat{\sigma}\left(\left|\frac{\Y{M}{k}-\E[\Y{M}{k}]}{\E[\Y{M}{k}]}\right|\right)/\sqrt{q}
    =\hat{\sigma}(\mathcal E)/\sqrt{q}\,.
\end{align*}

For a multivariate Gaussian distribution, the following corollaries give bounds on the coefficient of variation for the sample mean of log-minors and for the sample mean of subsystem entropy.

\begin{cor}[Coefficient of variation for a sample mean of log-minors] \label{cor:rel_err}
    Let $\lambda_1(M),\ldots,\lambda_n(M)$ be the eigenvalues of $M$; we order them from largest to smallest.
    Let $\ell(M) = \min\{\abs{\log \lambda_1(M)}, \abs{\log \lambda_n(M)}\}$.
    If $\ell(M)\neq0$, the coefficient of variation for a $q$-sample mean $S_Y$ of $\Y{M}{k}$ satisfies
    \begin{align}
        \cv{S_Y} \leq \frac{\log \hat{\kappa}}{\ell(M)}\sqrt{\frac{6(n-k)}{qkn}} \label{eq:cvy1}
    \end{align}
and
    \begin{align}
        \cv{S_Y} \leq \frac{\log \hat{\kappa}}{2\ell(M)}\sqrt{\frac{\wedge_{n,k}}{qk^2}}\,.\label{eq:cvy2}
    \end{align}
\end{cor}

\begin{proof}
    This corollary follows from \Cref{eq:cv_def}. We use \Cref{eq:ldi_varbound,eq:support_varbound} as upper bounds on the numerator. For all $k$, a lower bound on the denominator is $\abs{\E[Y_k(M)]} \geq k\ell(M)$.
\end{proof}

\begin{cor}[Coefficient of variation for mean subsystem entropy]
    For an $n$-variate Gaussian distribution with covariance matrix $M$, the coefficient of variation $c_v(S_h)$ for a $q$-sample mean $S_h$ of subsystem entropy satisfies
    \begin{align}
        c_v(S_h) \leq \frac{2\log \hat{\kappa}}{\ell(M) + \log 2e\pi}\sqrt{\frac{6(n-k)}{qkn}} \label{eq:cvh1}
    \end{align}
and
    \begin{align}
        c_v(S_h) \leq \frac{\log \hat{\kappa}}{\ell(M) + \log 2e\pi}\sqrt{\frac{\wedge_{n,k}}{qk^2}}\,.\label{eq:cvh2}
    \end{align}
\end{cor}

\begin{proof}
    We derive this result from \Cref{eq:cv_def}; we use \Cref{eq:ldi_varbound,eq:support_varbound} to bound the numerator, and we use \Cref{eq:diff_ent} to bound the expectation in the denominator.
\end{proof}

The bounds in \Cref{eq:cvy2,eq:cvh2} are sharper bounds than \Cref{eq:cvy1,eq:cvh1}. From \Cref{eq:cvy2,eq:cvh2}, we see that both $\cv{S_Y}$ and $\cv{S_h}$ decay in proportion to $\sqrt{k}$. Indeed, under a certain regularity condition (which we specify in \Cref{cor:rel_err_concentration}), the coefficient of variation decays to $0$ in the limit of large $n$ and large $k$.

\begin{cor}[Concentration of the relative error] \label{cor:rel_err_concentration}
    Let $\{M_i\}$ be a sequence of positive-definite matrices of dimension $n(i)$. Let $k = k(i) \leq n(i)$ be a function of $i$. Suppose that the sequence
    \begin{align}
        a_i := \sqrt{k(i)} \ell(M_i) \label{eq:criterion}
    \end{align}
    is nondecreasing and unbounded. It then follows that $c_v(S_Y) \rightarrow 0$ and $\mathcal{E}$ converges in probability to $0$ as $i$ becomes large.
\end{cor}

\begin{remark}
A sufficient condition for the concentration of $\mathcal E$ is that the sequence $\{M_i\}$ has fixed condition number and the smallest eigenvalue $\lambda_n$ is bounded away from both $0$ and $1$. Formally, the latter condition is
\begin{align}\label{eq:ccon}
    \text{there exists} \,\, \delta > 0 \,\, \text{such that, for all} \,\, i\,, \text{ we have}\,\, \lambda_{n(i)}(M_i) \in [\delta, 1 - \delta]\cap[1+\delta, \infty)\ \,.
\end{align}
\end{remark}

\begin{remark}
A popular model for sample covariance matrices is the Wishart ensemble\footnote{The Wishart ensemble $W_n(V,n_f)$ with scale matrix $V$ and $n_f$ degrees of freedom is the ensemble of random matrices $M:=n_f^{-1}\sum_{i=1}^{n_f}X_{i}^TX_{i}$, where the $X_1,X_2,\dots,X_{n_f}$ are $n_f$ realizations of an $n$-variate random variable with 0-mean Gaussian distribution $N_n(0,V)$ \cite{Goodman1963,Katzav2010}.}. A sequence $\{M_i\}$ of Wishart matrices can satisfy the condition in \Cref{eq:ccon} if the ratio $c:=n/n_f$ of the number $n$ of variables and the number $n_f$ of degrees of freedom is $c\notin\{1/4,1\}$ \cite{Katzav2010,Majumdar2010}.
\end{remark}

\begin{figure}[tp]
\centering
\includegraphics[trim={0cm 0.75cm 0cm 0.5cm},clip,width=1\textwidth]{fig_sampling}
\caption{Bounds on the standard error and coefficient of variation for a sample mean $S_Y$ of log-minors and a sample mean $S_h$ of subsystem entropy. In the left panels, we show the bounds B1 from \Cref{eq:se1} and B2 from \Cref{eq:se2} on the standard error of $S_Y$ and $S_h$. In the right panels, we show the bounds B1' (see \Cref{eq:cvy1,eq:cvh1}) and B2' (see \Cref{eq:cvy2,eq:cvh2}) on the coefficient of variation of $S_Y$ and $S_h$. We show results for progressively larger $n$ with fixed $k=30$ in panels \panel{A} and \panel{B}; results for progressively larger $k$ with fixed $n=400$
in panels \panel{C} and \panel{D}; and results for progressively larger $k$ with a fixed ratio $k/n=0.1$ in panels \panel{E} and \panel{F}. We compute bounds using $q=2000k$, $\K=3$, and $\ell(M)=1$.}
\label{fig:sampling_error}
\end{figure}

One can use these bounds on the standard error to choose a sample size $q$ that guarantees a desired accuracy of a sample mean. In Fig.\,\ref{fig:sampling_error}, we show our bounds on the standard error and the coefficient of variation of $S_Y$ and $S_h$ with $q=2000k$ and $\ell(M)=1$. In the left panels, we show the bounds B1 from \Cref{eq:se1} and B2 from \Cref{eq:se2} on the standard error of $S_Y$ and $S_h$. In the right panels, we show the bounds B1' (see \Cref{eq:cvy1,eq:cvh1}) and B2' (see \Cref{eq:cvy2,eq:cvh2}) on the coefficient of variation of $S_Y$ and $S_h$. In panels \panel{A} and \panel{B}, we vary the system size $n$ for fixed subsystem size $k$. We observe for $n\leq 2k$ that the values of the bounds B2 and B2' increase with $n$. For $n>2k$, the bounds B2 and B2' are independent of $n$. The bounds B1 and B1' are less sharp than the bounds B2 and B2'. The values of B1 and B1' increase with $n$ and approach their asymptotic values from below. For example, the bound B1 for $\hat\sigma(S_Y)$ has a limiting value of $\sqrt{6k/q}\times\log\K\approx0.06$. In panels \panel{C} and \panel{D}, we vary $k$ for fixed $n$. We observe for $k\leq n/2$ that the value of the bound B2 on the standard error is independent of $k$. For $k>n/2$, the value of B2 decreases with increasing $k$ and is $0$ for $k=n$. The bound B1 is less sharp than B2. Its value decreases with increasing $k$ for any $k\leq n$. The values of the bounds B1' and B2' on the coefficient of variation decrease with increasing subsystem size and are $0$ for $k=n$. In panels \panel{E} and \panel{F}, we vary $k$ for fixed ratio $k/n$, and we observe that the bounds on the standard error are independent of $k$ if the ratio $k/n$ is constant. The values of the bounds B1' and B2' decrease with increasing $k$. This is consistent with our previous observation that $\cv{S_Y}$ vanishes if the sequence $a_i$ (see \Cref{eq:criterion}) becomes unbounded.

It is important to note that all of our bounds on the standard error and the coefficient of variation of $S_Y$ and $S_h$ are asymptotically constant in $n$. It is thus not necessary to sample proportionally more minors from a larger matrix. Instead, to guarantee a desired accuracy of a sample mean of log-minors or subsystem entropy, one can choose $q$ to be a function of $k$. To ensure that the standard error is constant or decreases with growing $n$ and $k$, it is sufficient to choose $q$ in linear proportion to $k$. When the smallest and largest eigenvalues of a system's correlation matrix are fixed, one can ensure that the coefficient of variation is constant or decreasing with growing $n$ and $k$ by choosing $q$ in linear proportion to $k^{-1}$.

\section{Conclusions}\label{sec:conclusions}

We examined the problem of estimating the mean subsystem entropy of a system of $n$ coupled variables with covariance matrix $M$. When the joint distribution of a system's variables is an $n$-variate Gaussian, $t$, or Cauchy distribution, the mean differential entropy of subsystems is an affine function of the log-minors of the covariance matrix \cite{Ahmed1989,Nadarajah2005}. We derived tail and variance bounds on the distribution of log-minors of fixed size of a positive-definite matrix with bounded condition number. Using our variance bounds, we provided upper bounds on the standard error and on the coefficient of variation of both the sample mean of log-minors and the sample mean of subsystem entropy. Our results indicate that, despite the rapid growth of the number of subsystems with $n$, the accuracy of these sample means is asymptotically independent of a system's size. Instead, it is sufficient to increase the number of samples in linear proportion to the size of subsystems to achieve a desired sampling accuracy.

Our results are salient to studies that use mean subsystem entropy to examine systems of coupled variables \cite{Koetter2003,Tononi1994,Tononi1999}. Even for a system with as few as $50$ variables, sampling just 0.001\% of its subsystem entropies can require the computation of over a billion log-determinants. Using the largest and smallest eigenvalues of a system's covariance matrix to determine the number of samples that are needed to achieve a prescribed accuracy for a sample mean can thus facilitate a quantitative study of mean subsystem entropy when it would otherwise be impossible.

Throughout our paper, we relied only on knowledge of the largest and smallest eigenvalues of a system's covariance matrix. We expect that it is possible to derive sharper bounds than our current results when one knows the complete spectrum of a system's covariance matrix, likely by relying on Cauchy's interlacing theorem (\Cref{prop:cauchy}) to control the log-minors.

We presented two bounds on the variance of a log-minor that we choose uniformly at random from the set of log-minors of size $k$ of an $n\times n$ positive-definite matrix. The variance bound in \Cref{th:support} is sharper than the one
in \Cref{th:ldi}, but either bound is sufficient to deduce that the accuracy of a sample mean of subsystem entropy is asymptotically independent of a system's size and that one can achieve a prescribed accuracy by choosing the number of samples in linear proportion to the size of subsystems.

The proof of our first bound (see \Cref{sec:proof2}) relies on the existence of an upper bound for the difference between $\ld{A}$ for two different principal submatrices $A\in\mathcal A_k(M)$ and the invariance of $\ld{A}$ under a basis transformation of $A$. The proof of our second bound (see \Cref{sec:proof1}) relies on the existence of an upper bound and a lower bound for the support of the distribution $\p_{M,k}$.

Similar bounds and the invariance under basis transformation hold for several other matrix properties, including the largest and smallest eigenvalues. It is thus plausible that one can derive similar results for the standard error and coefficient of variation for many spectral properties of principal submatrices. For example, Chatterjee and Ledoux (2009) proved a large-deviation inequality for the empirical cumulative eigenvalue distribution of principal submatrices of Hermitian matrices \cite{Chatterjee2009}. These and other variance and tail bounds on submatrix properties offer welcoming  possibilities to enhance computational studies that characterize complex systems based on the mean properties of their subsystems. For example, they can provide guarantees for linear sketching techniques, which are relevant for data dimensionality reduction. They can also facilitate the use of methods of spectral graph analysis in the study of subgraphs, graphlets, and motifs in networks.

\section*{Acknowledgements}

We thank Cl\'ement Canonne, Kameron Decker Harris, Michael Neely, and participants of the \textit{IPAM Quantitative Linear Algebra Tutorials} for helpful discussions. A.C.S. was supported by the Clarendon Fund, e-Therapeutics plc, and funding from the Engineering and Physical Sciences Research Council under grant number EP/L016044/1. P.S.C. was supported by the National Science Foundation under Graduate Research Fellowship Grant 1122374.

redpajama_set_name: RedPajamaArXiv


We are an environmental non-government organisation (ENGO) working on a whole variety of issues – from farming to economics and from biodiversity to climate change. We work on these things because they are all linked. It is our objective to ensure that policy makers see those links and that strategies take note of the interconnected elements involved in policy decisions.
We started in 2002 as a group working on forestry. Since then we have grown to work in many different sectors; if you work with forestry issues, you have to look at soil, then water, then rivers, then coasts, then biodiversity and on and on. They are all systems and resources that our modern lives, although increasingly removed from nature, are dependent upon for survival.
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Q: pass parameters with post directly in the url without using forms I'm trying to pass parameters via post directly in the url, without using forms nor input buttons.
Not long ago I implemented a working solution, but now I'm reviewing my own the scripts and I can't have them do what I want. I don't remember having to tweak settings when the solution was working.
I give you a very simple example
<form method="post">
<name="x" />
</form>
<?php
print_r($_POST);
?>

If I call the script from the browser with
http://x.x.x.x/test.php?x=a

I can see that my parameter has not been passed.
Array ( )

I suspected a different behavior between php 5 and 7 but I tested both, also different browsers and 2 distros (linux and aws ami1).
Any hint? I still have the old scripts and they're all in this format, also I remember calling the url and the parameters in one line.

A: Any (properly encoded) data in the query string of the URL will be placed in $_GET by PHP.
$_POST will only contain properly encoded application/x-www-form-urlencoded or multipart/form-data data from the request body.
The only way to get any other kind of data into $_POST is to write it there with custom PHP code (and I can't think of any reason to do that other than for a quick and dirty backwards compatibility hack with pre-existing code that expects it in $_POST in a system that has changed the way it collects data).
(For "quick and dirty hack" read "you need to refactor this properly")

Asides:

*

*There is no <name> element in HTML

*Form data will only get to the request body if you put it there by submitting the form (or by copying it to a request made with Ajax).

A: I found out that I was actually NOT passing values in the url, they were in the body.
I don't get why Quentin says that "There is no  element in HTML" the script works fine, it was a mistake on the client I was using.

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I'm trying to get html of page which contains diacritics (í,č...). The problem is that urllib2.quote seems to not being work as I expected.
As far as I'm concerned, quote should convert url which contains diacritics to proper url.
Two things. First make sure you're including your shebang # -- coding: utf-8 -- at the top of your python script. This let's python know how to encode the text in your file. Second thing, you need to specify safe characters, which are not converted by the quote method. By default, only the / is specified as a safe character. This means that the : is being converted, which is breaking your URL.
So the first problem here is that urllib2's documentation is pretty poor. Going off the link that Kamal provided, I see no mention of the quote method in the docs. That makes trouble shooting pretty difficult.
With that said, let me explain this a little bit.
##   errors: specifies how errors are handled. Default is 'strict' which throws a UnicodeEncodeError, I think.
Not the answer you're looking for? Browse other questions tagged python html url urllib2 or ask your own question.
Why is my curl command working and not my python code?


Roopkund trek in India is one of the most popular trekking destinations among frequent trekkers. Roopkund trekking makes one aware about the beauty of deep virgin forests, gurgling brooks, breath-taking campsites, miles of undulating meadows, snow and ice and the taste of a great adventure as you climb from 8,000ft to 16,000ft. We start with dense forests bursting into Ali and Bedni Bugyal – India's most beautiful high altitude meadows. As we continue trekking higher and higher above Roopkund, we are only separated by air from the Trishul. Rock-strewn glaciers and snow clad mountains surround our journey. All along our trail, you will get to see mesmerizing views of peaks like Kedarnath, Chaukhamba, Neelkantha, Trishul & Nanda Ghunti. The difficulty level of the trek is high and to have the full best view get into this trek during May - June, and September – October.
Enjoy the breath-taking campsite of the gurgling brooks near the cascading water in the clean night sky under the billion stars.
Trek tail through the most alpine forest of Rhododendron and oak and other rare flora and fauna of the region.
Witness the mystery lake which is immersed and circled by hundreds of skeleton. This one major attraction with the skeletons of the pilgrims who fell to their deaths due to the unexpected natural calamities.
Enjoy the majestic view of Mt Trishul by taking some amazing photographs.
Admire the imposing Chaukhamba range, Neelkanth peak, Kedarnath, Kedar Dom.
We need to reach Kathgodam station which is one of the oldest and the central station in the district. It is well connected with other places like Lucknow, Delhi, Howrah and more.
From Delhi take the Ranikhet Express (5014) Dep. Delhi at 10:40 pm arrival Kathgodam at 5:30 am (Overnight Journey) or the Uttar sampark Kranti (5035) Dep. Delhi at 4:00pm arrival Kathgodam at 10:40pm (6 hrs.40min.Journey). Tickets from Delhi to Kathgodam by train cost around `150/- for a second class sleeper ticket and ` 400/- for a third A/C sleeper ticket. It is essential to book the tickets at least one month in advance as this route is always in demand.
Our trekking crew needs to be assembled at the Kathgodam Railway Station by 7.00am. From here we start our journey to the base camp Lohajing. We can notice the beauty of the narrow roads with curves, making the trekkers view the entire surrounding of lush green beauty jaw-dropping.
Bus stand backend trail takes us to the Raun Bagad. The walk is moderate difficulty with forests taking 2.5hrs to reach our destination of Raun Bagad. Raun Bagad is an iron bridge across the famous Neel Ganga River. Once you reach the bridge, we start with the climb to Didina. This is a steep climb for an hour and most of the time makes the trekkers tired soon. Once we have endured and overcome the tiredness and continue with the hill for about 1.5hrs. We will have an excellent mountaineering experience as we have finally reached our campsite for overnight stay and food.
Reaching our today's destination, we have two alternative routes. Take a look at the valley to witness a cluster of huts belonging to the shepherds of Tolpani village at a 3kms distance. From this village, we take a trail that climbs to Tolkaan in the northern direction. It makes us through a series of meadows switchbacks and lush oak mountains. It's time to get out of the forest by taking a sharp left towards the douth to reach Ali Bugyal.
The alternative route to reach Bedni Bugyal is to climb directly towards Bedni Bugyal without going through Toplani. Its 2.5jrs climb to reach Ali Bugyal.
We need to reach Ali Bugyal before afternoon to enjoy the seductive view. The trek is excellent before and after the monsoon. Have fun by clicking memories as we need to ascend further for 1.5 to 2hrs to reach Bedni Bugyal to reach our campsite for overnight stay and food.
Today is an easy day for all trekkers as we head towards Patap Navhauni. All our trekkers are requested to follow a straight line as we go to the other side of the ridge with a gradual increase in the slope. Morning view of Badani Bugyal is marvellous with green grass sounded view of Patti ka danda. Post breakfast we depart to Patar Nachauni which takes max 2-3hrs to reach. Enjoy the best picture with mountain, Waterfalls, small-small water lakes between meadows.
We start making a steep dash to gain altitude as we trek in the snow line. Be prepared to feel the effects of weather change as it is a high altitude trekking. To acclimatize to the weather, we start our trek early to reach Bhawabasa. We take the steep zigzag route to climb the Kalu Vinayak Mountain for around 1.5hrs. Bhagwabasa camp is gradual slope downwards and is 2kms away from the view of Bhagwabasa huts.
Outstanding day of the trek and so we start the day early at 4 am. The early start is due to the snow which is hard now and turns soft as the sun rises. We are 3kms away from Roopkund, which is an uphill climb from here. The trail takes nearly 3-3.5hrs to reach Roopkund. Climb Roopkund, and we must climb to Junargali. Junargali is a must visit for trekkers unless the weather is too harsh. Once we enjoy the beauty of the sight, we start early to return to Bedni Bugyal. Be careful with every step as the snow slopes are tricky.
Another great trek of the complete trek package. With a simple path filled with oak forests and scenic beauty to see the end of meadows. On a clear day, you would be astonished by looking at the Trishul on your way back from Bedni Bugyal to Wan. Get down from the Rhododendron forest with a steep descent to the Neel Ganga. From here we are just 30mins climb away from Wan village. From here we take an hour drive back to Lohajung Basecamp.
Lohajung base in itself is a beautiful place to relax and chill after the high altitude trek. Don't think more; we got this day covered. Enjoy the weather as our fellow representatives take you back to Kathgodam at the day's end.
Best time to go for Roopkund Trek is during May - June, and September – October.
2. How difficult is the Roopkund Trek?
Roopkund Trek in comparison is Moderate to Difficult difficulty with another high altitude trek.
On an average, we cover roughly 4-7kms between our camping stations.


In 2004, the Illinois Supreme Court looked at the hybrid child representative role and found it unconstitutional as applied in In re Marriage of Bates.230 In the Bates trial court, the mother asked that the child representative be required to testify or, in the alternative, that his report be stricken.231 Her motion also asked that the child representative statute be declared unconstitutional because it allowed the child representative to make recommendations without allowing the parties to cross examine.232 The trial court denied the motion, admitted the report as evidence, and relied on the child representative's report in making custody, parenting time and support determinations ( although it was consistent with the trial testimony as well.)233 The court of appeals held that the child representative could make recommendations based on personal observations and could be questioned about those observations as a witness; thus, the opportunity evaluation of the best interests of the child); 750 ILL. COMP. STAT. 5/604.10(d)(West 2017) (the court may order an investigation and report to assist in allocating parental responsibilities.) Illinois Supreme Court Rule 215 also allows for mental and physical examination of the parties and others persons in the event that their mental health is at issue in the litigation.
227 750 ILL. COMP. STAT. 5/506.
228 See Rebecca J. Whitcombe, The Child's Representative Law After In Re Marriage of Bates, 93 ILL. B. J. 26 (2005) ("For years, only two types of representatives were available - the guardian ad litem (GAL) and the attorney for the child. . . . That changed effective 2000 with the enactment of an amendment to section 506 creating the child's representative, an attorney-for-the-child/GAL hybrid designed to combine the best features of both."(internal citations omitted). 229 See Carl W. Gilmore, Understanding the Illinois Child's Representative Statute, 89 ILL. B. J. 458 (2001). 230In re Marriage of Bates, 819 N.E.2d 714, 728 (Ill. App. Ct. 2004). 231 Id. 232 Id. 233 In re Marriage of Bates, 819 N.E.2d at 729.


We do not only build the finest brass instruments, we also upgrade instruments, which were not built by us. It doesn't matter if it's a trumpet from Bach, a Conn trombone or if you own and love a Yamaha instrument – if you think it can be improved, do not hesitate and come to us. Is it the bell, the corpus, a tuning slide or a lead pipe finished with Singing Bell technique – your instrument, your ears, the ears of your audience and maybe the ears of your audio engineer will thank you.
You simply send your Instrument as a whole or just the part you want to be upgraded and we will do the job! And if you do not want us to change the original part, we also offer completely new parts finished with the Singing Bell technique – exactly as you wish or an improved copy of a draft.
A trumpet bell modified with the Singing Bell techniqe (depending on the surface) will cost from 650 EUR, the lead pipe of a trumpet already starting at 190 EUR. If you want to know the exact price, do not hesitate to contact us.
This short photo story should give you some idea about the different steps of the upgrading process of a Yamaha YTR-6335HII.


In the cold rolling process, a hot-rolled coil is made thinner by passing it between two rolls at room temperature. Not only is the thickness of the material made uniform, but the internal structure is made homogeneous at a microscopic level, and the surface of the roll is transferred to the material to provide a polished finish. To eliminate warping that may harden and to achieve the desired properties, the product is then heat treated during the intermediate and post processing steps.
Our three-group 20-high Sendzimir cold rolling mill handles a diverse range of rolling demands.
We own five annealing and pickling lines with the diversity to manufacture various product types and shapes under a broad range of conditions.
This flexibility allows us to fine tune our response to different production demands.


Talking Babies, Violin Playing Beavers, Shrunken Linebackers…And a Half Naked Megan Fox.
I didn't watch the superbowl last year. I was in México, a country that treated the occasion like any other Sunday. I did as well, by drinking tequila and joya in my hammock and being as lazy as possible. Long story short, I eventually figured out that the superbowl was on, turned on my TV, and then watched until I figured out that the same commercials didn't air at which point I turned the game off and continued drinking in my hammock.
The Doritos Series – These 4 ads were some of my favorites. I had previously seen the coffin ad online, but the baby/dad on a date and the Doritos samurai were pretty solid. Something resonated with me about your crazy friend who really would make a samurai suit out of Doritos chips. Matt Lewis? I thought the barking collar ad was particularly funny. I speak to this out of experience. That's right, when I was about 12, my 14 year old brother convinced me to give our dog Sandy's barking collar a try on my own vocal chords. Long story short, after the meager-est of utterances, I can now say with confidence that those things bring the ruckus to one's pipes – don't put them on your dogs, or worse, let your dog put one on you.
I love the E*trade babies. Milk-a-holic. Get it? She's a baby, so in stead of being an alcoholic she's a milk a holic. Hahahahhaaa. And she's napping around… Brilliant. As the great former OL wrangler Danny once sagely stated "I don't like babies…they can't hold a conversation." The inherent truth in results in that there is something about an articulate, well spoken infant that makes you listen to every word he's saying. Maybe it's that for me, this paints a picture of a perfect world wherein I could like small children, but in the present I must maintain as a general policy not fraternizing with things that I can't reason or communicate with. Where can I adopt one of these e*trade kids?
Brett Farve, MVP 2020 – The best bits of humor are based in truth and actuality. This ad featured Brett Farve accepting the 2020 superbowl MVP award and talking about whether or not he would be returning next season (note: this would be happening currently in a post-game press conference if it weren't for that interception…). Will Brett Farve be coming back next year? I don't really care, but I look forward to and welcome the interruptions of the never-ending stream of baseball highlights on Sportscenter all this summer.
Honorable mentions go out to the beaver who plays violin (funny by default – its a beaver. Who plays the violin…), the kid who let's his dad fall off the ladder because of Megan Fox's multimedia message (I've almost done this myself, minus the Megan Fox. Sorry dad.), the Kayne/T-Pain style voice morphing bud light commercial (It was a LIttLLllEE LaaaAteEEE, but still applicable) the Punxsutawney Palamoau (shrunken celebs…always funny), and the hangover-style commercial about getting the whale out of the back of the Escalade, which reminds me of my night last night, minus the whale. And the Escalade.
Much like his soon to be pro career, the Tim Tebow ad attracted a lot of hype and delivered nothing particularly controversial or moving. I am, however, disappointed in CBS's decision to politicize, but look forward to this response that will be run next year by the pro-choice movement.
I would also like to mention that this is the first football I've been able to watch this year without mistakenly thinking that the two teams in competition were Verison and at&t's 3G maps.
And to conclude, I say that this video of the "Who Dat Dog" should have been shown during a commercial break just because it is awesome. Take that thing to Bourbon street tonight, he'll have the time of his life.


text:
Music ›
Vanessa Carlton talks life after "A Thousand Miles"
Published on January 29, 2016 at 12:08 am Last update on August 3, 2017 at 5:52 pm
Singer-songwriter Vanessa Carlton, who catapulted to fame in 2001 with "A Thousand Miles" and "Ordinary Day," will perform at the Parish on Saturday. Carlton caught up with The Daily Texan to talk about touring, motherhood and her new album, Liberman:
The Daily Texan: How's your tour going?
Vanessa Carlton: It's been probably our best tour so far. I think the shows have been really good. We mostly play Liberman, and people seem to really connect to the new record, so it feels good.
DT: This album is more mellow than much of your previous work. What inspired that choice?
VC: We wanted [the album] to feel dreamy, almost watery. I think working with [producer] Steve Osborne was a huge part of that, because working with his aesthetic is kind of eerie and beautiful. It's not an autobiographical record by any means and I wanted the sounds to match that.
DT: How long did it take for Liberman to come together?
VC: It was tricky in the beginning to get the sonic palette right. We tried a few different things, but once we got it, it was the best feeling. We recorded the majority of it in England, so I had to go over there a couple times before we got it right. It was just [Osborne] and I in the studio, so it was pretty intense.
DT: How has motherhood impacted your music or how you go about planning a tour?
VC: The majority of the album was written and done before I got pregnant, but having a baby on the road is a whole new game. I think this has been a total experiment to see if I could do it, and it's kind of working, which is great because it means I can perform around the country and still keep my family together. From here on out, I'll probably only go out for three-week spurts and then go home. The other way [my daughter] really affected me is in the performance every night. When you become a mom, there's something about it that makes you very much in the moment. Your baby doesn't care about your stupid problems, they're just these pure little beings that want to connect with you.
DT: Initially, how did you approach writing new material after "A Thousand Miles?" Was there added pressure to write another hit?
VC: Honestly the way I look at that is I kind of pretend it didn't happen. This tour is not a nostalgia tour for us — we play mostly new material. ["A Thousand Miles"] was almost a different lifetime ago. I understand people know that song and love that song, but if they're still connecting with me now, it's not really about that song anymore. The people who do come to the show just to hear that song often end up connecting to the new stuff, and that's good. I feel no obligation to fulfill anyone's expectations from that earlier era. I'm 35 years old. I've grown up. I've moved forward.
DT: How do you think location, whether it's recording in England or moving to Nashville, impacted this record?
VC: Working in England is really incredible, because you're kind of in the middle of nowhere. It's very isolating, in a good way, for an artist. You live like a monk — all you do every day is work on your project. There's something somber about the record, and I think that really reflects the tone of where we were. Rainy England is a feeling on this album. Nashville is a different thing. We played anything we could get our hands on to create the tracks and to cut the songs in a way that really linked in with what I'd done in England. I think your environment absolutely affects the kind of music you're making, at least to a certain degree.
DT: Is there a song you're most proud of or that was particularly difficult to write?
VC: I love the story of "Matter of Time," because it was such a quick and natural recording. and my husband plays with me. I had written the song on piano, and it just didn't sound right. I asked him to play guitar, and he did this finger picking that was really beautiful. It just made the song. When I started singing over it, it just sounded so good. We ended up taking the first take. It was one of those magic moments in the studio.
Who: Vanessa Carlton
Where: Parish
When: Saturday, Jan. 30

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The future of Pterofin is looking bright
I woke up Friday morning thinking about preparing for a giant storm to hit Seattle while also planning to hand deliver a few of my wind prototypes to a large ballroom at the University of Washington for a career day event. They were expecting two to three hundred students to show up with resumes in hand, however I was told that more than 500 students came through to apply for summer internships at almost two dozen companies.
I was honored to be in the presence of recruiters from large companies such as Boeing, Microsoft, and the US Navy, but even more honored to be handed almost a hundred resumes from some of the most ambitious electrical and aeronautical engineering students I've ever met. Most of them already studied my website and had some great questions thought up inquiring about me and the future of Pterofin, Inc.
Needless to say I had a few incredible discussions with some very eager students who wanted to spend their time interning and helping develop Pterofin technology. Some of them were extremely passionate about the usefulness of the technology which helps deliver clean drinking water to populations who fight that battle day after day and year after year. Other students were very animated and passionate about the physics behind Pterofin technology, and how they'd absolutely love to be a part of my team doing R&D.
I can't even begin to tell you how excited I am about meeting some of the future leaders in technology, taking such an active interest in clean energy and playing their part to not only help save our environment, but also save lives on a global scale. I'd like to think they're all keeping their fingers crossed and hoping I'll get back to them before the fist day of their summer vacation, as I think that's how long it might take me to go through the stack of resumes and make my decision.
Pterofin proceeding on painstaking patent process
The process of seeking a patent is anything but quick. The wheels of government turn slowly, if it all, and the patent process is no different.
We've hoped that the path to earning a patent award of our Pterofin designs would be quicker, but it's been a time and resource-consuming process of back and forth between us, our patent attorneys and and the US Patent Office. Fortunately, a just-announced program will hopefully finally help us achieve that goal.
The government recently debuted a pilot program called P3. While the nomenclature may not be inspired, program itself is in that it allows entrepreneurs a little more leeway in navigating the patent process. It gives us an avenue where we can actually interact in real-time with officials, answer their questions, have our own questions answered, and hopefully clear any hurdles and get that final approval we've been seeking for so long.
We appreciate the supporters who have joined us on the journey as we try to bring a clean energy source to everyone. As we continue to pay the ever-mounting legal fees that this process generates, we ask that you share our plans with other like-minded individuals who believe in simple solutions to help people and the planet. Together we can change the world.
Pterofin taking part in TedX event
Ted and TedX talks are renowned for bringing in some of the best minds in the world to discuss all sorts of topics that are life and world changing. This year's TedX San Juan Island event is no different, as some fantastic speakers will gather September 17 to discus quality of life.
Quality of life is a topic at the forefront of what we do, as the technology we're working on developing is intended to improve life for people throughout the world. From someone in American looking to bolster their self sufficiency to someone in the developing world trying to gain access to clean water, our simple designs are meant to make life better.
One of the speakers is our own Jarrod Lyman, a recent addition to the Pterofin team who is helping with social media. We don't want to spoil the speech for you, but his talk will focus on many issues that we are striving to address.
One of the most difficult parts of creating a new company is spreading the word about your vision and getting others to follow along on that journey. Wallace has an inspiring vision, but Pterofin isn't the vision, it's the vehicle that drives us to the scenic viewpoint that overlooks that vision: a world where everyone has access to clean water and efficiently generated electricity. The TedX event will allow us to bring more people along on that trip.
Berkeley students pursue continued success with Pterofin
In October of 2015 I was approached by a team of students from UC Berkeley who were working on an oscillating wind power device that would be sited in Marin County, California. After performing some background research in the field they found my company, Pterofin, Inc., and were highly impressed with the work I've done, saying it was, "particularly inspiring and actually aligns with some of the preliminary designs [they] have been exploring". They said they'd love to learn more about my products and look for any kind of opportunity to collaborate in scaling up my technology and collect data on it's efficiency. Needless to say, I was flattered and honored to work with such an eager and distinguished group of students who were not only impressed with the work I've done, but also excited about exploring my technology and an alternative source of renewable energy.
After signing a NDA and creating a Memorandum of Understanding that we all agreed upon, I began advising the group of students who later mentioned, "The establishment of collaborating with Wallace Kempkey, founder of Pterofin, was instrumental in the advancement of our project from a stage with rough prototypes of multiple concepts to one with a higher-fidelity functional prototype of a single concept." I told them about the dozens of prototypes I've made and tested in the wind and water, and everything I learned from observing and tweaking them. One of the students immediately caught a flight up to Seattle to meet me and have me talk through a prototype they were particularly interested in replicating. I exchanged dozens of emails and spent many hours over the phone working out solutions for designing, building and testing their final project.
Last week the student team shared all of the information they had accumulated from their research and invited me to continue working with next years students who will be picking up where they left off. One of the students mentioned in their final paper, "If, after this review and exploration, the team decides to continue working on the technology we developed with the help of Wallace from Pterofin, then I suggest they continue optimizing the device for electricity generation."
Meeting the energy demands of our future generations is going to be a monumental challenge and can dramatically improve the quality of life of potentially billions of people. Whether it's pumping clean drinking water to the surface or powering lights and refrigerating food or vaccinations, human kind could benefit from our research and knowledge in understanding the forces of our natural environment. Through knowledge we can adapt, persevere and flourish as a species on this gift to humankind we call Earth.
Pterofin Announces Incorporation!
Imagine a world where there is no bill for electricity.
Instead a fin dips water or a wing tips wind to turn the earth's natural currents into power to pump water and provide electricity for your home.
Click image to watch video of latest prototypes!
What it Takes to Make It
To make it in the world of renewable energy technology, it isn't just about hard work, engineering design, and testing the prototypes you create. It's also about protecting your idea, your business, and securing investment.
In August of last year Pterofin filed an application for a non-provisional patent and qualified for a green fast-track patent under President Obama's American Recovery and Reinvestment Act. Pterofin's patent is expected in 2013. And this month, Pterofin took another step forward to secure its business future.
Pterofin is now officially incorporated which provides the company with financial and legal protections to enter a new phase of growth. The journey to success and design manufacture is that much closer.
Pterofin is ramping up for the Northwest Regional Finals of the Clean Tech Open taking place this October. For the taking is $250K, and the Pterofin team is focused and determined.
Pterofin is also at a critical crossroad, requiring us to reach out and facilitate the investment required to continue pursuit of a patent.
Pterofin's goal is to raise $200K by September. This fundraising will secure the software we need to validate the technology's design in addition to payment to lawyers pushing our patent paperwork through the US Patent Office.
Asia? Africa?
Wally, CEO of Pterofin, spent this summer touring the Pacific NW and California, meeting with Hong Kong Trade Groups, US trade leaders, and National Laboratories. He is exploring opportunity in Asia and in touch with many representatives that can protect a venture into global markets.
Clinton, COO of Pterofin, is developing potential partnerships with distribution leaders in Africa, a continent with strong appetite for energy.
Latest in Pterofin design
Since testing the Pterofin prototype this past March at the Kirsten Wind Tunnel and Water Tunnel at University of Washington, the team has been fine-tuning and fast-tracking design validations, pushing the technology to the limit.
The Pterofin prototype is inspired by natural movements in the flapping of a bird's wing. The biomimicry evident in the design, takes a natural approach to water pump technology. Opportunities for deployment include regions of the world where the wind speed is low and water resources cannot be piped.
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Wallace Kempkey, Jarrod Lyman
Pterofin


The Buckhorn Café offers everything from traditional Texas BBQ brisket, ribs and chicken to mouth-watering bacon & cheddar burgers. Try the Buckhorn Burger - it's a local favorite! The Café also has fresh salads and several menu items for the kids. The Buckhorn Café is open daily for lunch from 11:00 am until 3:00 pm. Extended hours Memorial Day through Labor Day.
If you have more than 20 in your group, call ahead for discount group lunch rates to our group reservations office.
Check out our museum while you are here!
The Buckhorn Museum features wildlife from all over the world; including fish from the seven seas, animals from every continent and strange animal oddities which have been collected for over 100 years! Our exhibit halls are comprised of over 520 species of wildlife—many of which are record holders.
The Buckhorn's most prized record holder is the famous '78 Point Buck'. This impressive rack was sold to the saloon owner himself, Albert Friedrich, for a mere $100 dollars back in 1890. Other amazing specimens include a record 1,056 pound black marlin and a 10,000 year old pre-historic Irish elk.
In 1881, seventeen-year-old Albert Friedrich, a bartender and bellhop at San Antonio's Southern Hotel on Main Plaza, decided to open his own saloon across the street. Little did Friedrich know, over 130 years later his Buckhorn Saloon and Museum would be one of San Antonio's top attractions and continue to entertain tourists from all over the world.
The Buckhorn Saloon and Museum has been enjoyed for more than a century, and with the continuation and expansion of the Buckhorn Museum, it will continue to delight patrons for generations to come.


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Our wellbeing and the future of our children depend CSU graduates who can help ensure we continue to have access to basic life sustaining resources and that our and the health of the planet is protected. As nations look to reduce carbon emissions, mitigate against extreme weather events, develop technologies of the future, and ensure future generations have the same opportunities we currently enjoy, sustainable practices are becoming increasingly important. Many businesses, governments and organisations are looking for graduates who can help facilitate this transition.
For example, law and policing students can apply sustainability when considering social equity, justice, citizenship and biosecurity.
Investigation of evidence about environmental, social, and economic considerations on a contextualised topic.
Ensure assessments explore the tension in the complexities of sustainable decision-making.
At least one assessment item requiring sustainable decision making in a professional context with consideration of ecological consequences.
Case studies, discussion, debates and group presentations on issues associated with internationalisation, ethical practice, equality, health, climate change, planning and development, resource use, diversity, and biodiversity are all part of sustainable practice.
Assessment of this GLO needs to be explicit and contextualised. The following example rubric is for a group of students who are asked to demonstrate an understanding of the 'knowledge' element of sustainable for some form of professional practice.
This addresses both the sustainable practice and professional practice GLO.
Save this document as a pdf.


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MTW Puts People First
This fast-growing outfit creates major e-commerce applications. But its core product is its culture. "We thought that if we started by creating a place where people liked to work, we'd achieve our financial goals."
By Jill Rosenfeld 4 minute Read
Life as an IT consultant can be financially rewarding, even intellectually challenging. It's just not very much of a, well, life. It's not unusual for an IT consultant to be on the road for four days every week, writing code and installing systems, and to stay late on the few days that he or she is in the home office. Why do so many smart people put up with such a punishing lifestyle? It's the price you have to pay for being part of a fast-growing professional-services industry.
MTW Corp. isn't willing to pay that price. The company, which is headquartered outside of Kansas City, provides software and Internet applications to the financial-services industry and state governments. It has an enviable track record. Since 1996, when Ed Ossie, 45, president and COO, left Texas Instruments to join MTW, it has expanded from a 50-person outfit with $8 million in revenues to a 200-person powerhouse with $31 million in revenues.
The company has an even more enviable record of challenging conventional wisdom on the human side of the enterprise. MTW takes its people, its intellectual capital, and its evolving culture seriously. "Task teams" make many of the companywide operational decisions. Its technical advisory council, which sets the technology agenda, includes two elected members. MTW has built seven software-development centers around the country that serve as home bases for its employees. These regional centers make it possible for clients to come to MTW, rather than the other way around, which saves employees from the wear and tear of travel.
"We thought that if we started by creating a place where people liked to work, we'd achieve our financial goals," says Richard Mueller, 55, MTW's founder and CEO.
It's a commonsense insight that even the smartest professional-services companies tend to ignore. One of MTW's key tools is an "expectation agreement" that establishes a set of mutually agreed-on objectives between an individual, his or her boss, and the team. The agreement gets updated every six months. An expectation can be as simple as getting one day off a week to care for an ailing parent. "Usually, people feel they need to develop credibility before sharing that they have a particular challenge in their lives," says Mueller. "We've found that if we give people the benefit of the doubt and draw them out, we rarely suffer because of it."
Where MTW people work is as important as how they work. Most MTW employees are distributed among the company's seven regional offices. Employees like that arrangement because they can work near their homes and can also participate in MTW decisions on a day-to-day basis. (After all, on-the-go consultants spend far more time in their clients' offices than they do working with their own colleagues.)
And participate, they do. Employee groups make many of the company's core operational decisions. For example, a technical advisory council explores emerging technologies and looks at how those technologies might be applied to various markets. The council also develops standards and documents best-practices guidelines. Headed by MTW's chief technologist, the council originally had six appointed members, until those members decided to add two elected seats. "We added those two seats because we wanted to create a democratic process," says software designer Rao Mulpuri, 33. Each council member is in charge of a subgroup that meets bimonthly on a specific market or technology.
Decisions do take longer to accomplish by committee, Ossie concedes. "But the buy-in, trust, and confidence in the outcome is well worth the time," he says. "And for me, on the back end, the process is simple: I provide the stamp of approval."
Aaron Shumate, 27, approves too. He left Andersen Consulting to work for MTW as a business-development manager. He realized he needed to make a change when he found himself, night after night, longing for home while inhabiting a corporate apartment in Santa Fe, New Mexico. Now he works in the Kansas City office, closer to his family. But "the biggest difference is the amount of responsibility and the voice I'm given in the decision-making process," he says. "That ownership gives me energy at work every day."
Contact Richard Mueller by email (rmueller@mtwconsulting.com), or visit MTW on the Web (www.mtwconsulting.com).
Sidebar: Culture Club
Of the many teams that help to make decisions for MTW Corp., only one — the Culture and Communications Team — reserves a chair for founder and CEO Richard Mueller. President Ed Ossie is also an active participant. Why? "Because if we get the culture right," says Ossie, "then that drives the whole company."
The group has 12 members and meets by conference call at least once a month, though it often meets more frequently. One recent project was a companywide survey of "guiding principles." The team was charged with identifying a dozen principles, along with a list of questions for evaluating how well MTW was living up to them. Senior systems analyst Chayan Dasgupta, 32, came up with his questions by asking coworkers for suggestions.
The survey asked employees to rank 12 principles — trust, integrity, customer satisfaction, and so on — by importance, and then to rate the company's adherence to each one. Employees ranked "learning organization" as one of the top three, but indicated that MTW was failing to live up to its goal by a wide margin.
Results were posted on MTW's intranet, frank criticism and all. MTW doesn't claim to be perfect, but it does try to give employees a hand in addressing imperfections. "People recognize that this is an evolving endeavor," says Mueller. "And they appreciate being part of the process."


Make it for a party appetizer and eat the leftovers the next day. The slaw will last in the fridge for 2-3 days.
Toast almonds and coconut shreds on the stove for a few minutes.
Cut the cabbage into thin slices.
Combine all the ingredients and toss with the dressing.
Thanks for giving me coleslaw inspiration.