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	<title>An Ergodic Walk</title>
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	<description>a process whose average over time converges to the true average</description>
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		<title>CFP : GlobalSIP 2013 Deadline Extended to June 15</title>
		<link>http://ergodicity.net/2013/05/17/cfp-globalsip-2013-deadline-extended-to-june-15/</link>
		<comments>http://ergodicity.net/2013/05/17/cfp-globalsip-2013-deadline-extended-to-june-15/#comments</comments>
		<pubDate>Fri, 17 May 2013 17:26:18 +0000</pubDate>
		<dc:creator>Anand Sarwate</dc:creator>
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		<category><![CDATA[conferences]]></category>
		<category><![CDATA[privacy]]></category>
		<category><![CDATA[signal processing]]></category>

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		<description><![CDATA[I&#8217;m on the program committee for the Cyber-Security and Privacy symposium, so I figured I would post this here to make more work for myself. GlobalSIP 2013 &#8211; Call for Papers IEEE Global Conference on Signal and Information Processing December 3-5, 2013 &#124; Austin, Texas, U.S.A. GlobalSIP: IEEE Global Conference on Signal and Information Processing [&#8230;]<img alt="" border="0" src="http://stats.wordpress.com/b.gif?host=ergodicity.net&#038;blog=1934944&#038;post=3698&#038;subd=ergodicwalk&#038;ref=&#038;feed=1" width="1" height="1" />]]></description>
				<content:encoded><![CDATA[<p>I&#8217;m on the program committee for the <a href="http://www.ieeeglobalsip.org/sym/13/csp">Cyber-Security and Privacy</a> symposium, so I figured I would post this here to make more work for myself.</p>
<p>GlobalSIP 2013 &#8211; Call for Papers<br />
<a href="http://www.ieeeglobalsip.org">IEEE Global Conference on Signal and Information Processing</a><br />
December 3-5, 2013 | Austin, Texas, U.S.A.</p>
<p><a href="http://www.ieeeglobalsip.org">GlobalSIP: IEEE Global Conference on Signal and Information Processing</a> is a new flagship IEEE Signal Processing Society conference. The focus of this conference is on signal and information processing and up-and-coming signal processing themes.</p>
<p>GlobalSIP is composed of symposia selected based on responses to the call-for-symposia proposals. GlobalSIP is composed of symposia on hot topics related to signal and information processing.</p>
<p>The selected symposia are:</p>
<ul>
<li><a href="http://www.ieeeglobalsip.org/sym/13/anebd">Advancing Neural Engineering Through Big Data</a></li>
<li><a href="http://www.ieeeglobalsip.org/sym/13/bsb">Bioinformatics and Systems Biology</a></li>
<li><a href="http://www.ieeeglobalsip.org/sym/13/csi">Controlled Sensing For Inference: Applications, Theory and Algorithms</a></li>
<li><a href="http://www.ieeeglobalsip.org/sym/13/csp">Cyber-Security and Privacy</a></li>
<li><a href="http://www.ieeeglobalsip.org/sym/13/ecnsic">Emerging Challenges in Network Sensing, Inference, and Communication</a></li>
<li><a href="http://www.ieeeglobalsip.org/sym/13/ehgwc">Energy Harvesting and Green Wireless Communications</a></li>
<li><a href="http://www.ieeeglobalsip.org/sym/13/gsp">Graph Signal Processing</a></li>
<li><a href="http://www.ieeeglobalsip.org/sym/13/ipsg">Information Processing in the Smart Grid</a></li>
<li><a href="http://www.ieeeglobalsip.org/sym/13/ipn">Information Processing over Networks</a></li>
<li><a href="http://www.ieeeglobalsip.org/sym/13/ldmosp">Low-Dimensional Models and Optimization in Signal Processing</a></li>
<li><a href="http://www.ieeeglobalsip.org/sym/13/lpssp">Low-Power Systems and Signal Processing</a></li>
<li><a href="http://www.ieeeglobalsip.org/sym/13/mmwis">Millimeter Wave Imaging and Sensing</a></li>
<li><a href="http://www.ieeeglobalsip.org/sym/13/mi">Mobile Imaging</a></li>
<li><a href="http://www.ieeeglobalsip.org/sym/13/nt">Network Theory</a></li>
<li><a href="http://www.ieeeglobalsip.org/sym/13/nssim">New Sensing and Statistical Inference Methods</a></li>
<li><a href="http://www.ieeeglobalsip.org/sym/13/omlsp">Optimization in Machine Learning and Signal Processing</a></li>
<li><a href="http://www.ieeeglobalsip.org/sym/13/sipfe">Signal and Information Processing in Finance and Economics</a></li>
<li><a href="http://www.ieeeglobalsip.org/sym/13/sdcr">Software Defined and Cognitive Radios</a></li>
</ul>
<p>Paper submission will be online only through the <a href="http://www.ieeeglobalsip.org/">GlobalSIP 2013 website</a> Papers should be in IEEE two-column format. The maximum length varies among the symposia; be sure to check each symposium’s information page for details. Authors of Signal Processing Letters papers will be given the opportunity to present their work at GlobalSIP 2013, subject to space availability and approval by the Technical Program Chairs of GlobalSIP 2013. The authors need to specify in which symposium they wish to present their paper. Please check conference webpage for details.</p>
<p>Important Dates:<br />
*New* Paper Submission Deadline &#8211; June 15, 2013<br />
Review Results Announce &#8211; July 30, 2013<br />
Camera-Ready Papers Due &#8211; September 7, 2013<br />
*New* SPL request for presentation &#8211; September 7, 2013</p>
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		<title>Persi Diaconis on coincidence</title>
		<link>http://ergodicity.net/2013/05/03/persi-diaconis-on-coincidence/</link>
		<comments>http://ergodicity.net/2013/05/03/persi-diaconis-on-coincidence/#comments</comments>
		<pubDate>Fri, 03 May 2013 19:48:01 +0000</pubDate>
		<dc:creator>Anand Sarwate</dc:creator>
				<category><![CDATA[Uncategorized]]></category>
		<category><![CDATA[probability]]></category>
		<category><![CDATA[statistics]]></category>

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		<description><![CDATA[Persi Diaconis gave the second annual Billingsley Lecture at UChicago yesterday on the topic of coincidences and what a skeptical statistician/probabilist should say about them. He started out by talking about how Jung was fascinated by paradoxes (apparently there&#8217;s one about having fish come up all the time in conversation). It was mostly a general-audience [&#8230;]<img alt="" border="0" src="http://stats.wordpress.com/b.gif?host=ergodicity.net&#038;blog=1934944&#038;post=3693&#038;subd=ergodicwalk&#038;ref=&#038;feed=1" width="1" height="1" />]]></description>
				<content:encoded><![CDATA[<p><a href="http://en.wikipedia.org/wiki/Persi_Diaconis">Persi Diaconis</a> gave the second annual <a href="http://statistics.uchicago.edu/seminars/billingsley/">Billingsley Lecture</a> at UChicago yesterday on the <a href="http://www.tandfonline.com/doi/abs/10.1080/01621459.1989.10478847">topic of coincidences</a> and what a skeptical statistician/probabilist should say about them.  He started out by talking about how Jung was fascinated by paradoxes (apparently there&#8217;s one about having fish come up all the time in conversation).</p>
<p>It was mostly a general-audience talk (with some asides about Poisson approximation), and the first part on the <a href="https://en.wikipedia.org/wiki/Birthday_problem">birthday problem</a> and variants.  Abstracted away, the question is given <img src='http://s0.wp.com/latex.php?latex=n&amp;bg=ffffff&amp;fg=29303b&amp;s=0' alt='n' title='n' class='latex' /> balls (people) and <img src='http://s0.wp.com/latex.php?latex=C&amp;bg=ffffff&amp;fg=29303b&amp;s=0' alt='C' title='C' class='latex' /> bins/categories (days), how big should <img src='http://s0.wp.com/latex.php?latex=n&amp;bg=ffffff&amp;fg=29303b&amp;s=0' alt='n' title='n' class='latex' /> be so that there&#8217;s an even chance that two balls land in the same bin?  Turns out <img src='http://s0.wp.com/latex.php?latex=n+%5Capprox+latex+1.2+%5Csqrt%7BC%7D&amp;bg=ffffff&amp;fg=29303b&amp;s=0' alt='n &#92;approx latex 1.2 &#92;sqrt{C}' title='n &#92;approx latex 1.2 &#92;sqrt{C}' class='latex' />, as we know, but we can expand this to deal with approximate matches (you need only 7 people to get 2 birthdays in the same week with probability around 1/2).  If you want to put a graph on it you can ask social-network coincidence questions and get some scalings as a function of the number of edges and number of categories &#8212; here there are <img src='http://s0.wp.com/latex.php?latex=n&amp;bg=ffffff&amp;fg=29303b&amp;s=0' alt='n' title='n' class='latex' /> vertices and <img src='http://s0.wp.com/latex.php?latex=C&amp;bg=ffffff&amp;fg=29303b&amp;s=0' alt='C' title='C' class='latex' /> colors for the vertices.  What these calculations show, of course, is that most coincidences are not so surprising, at least in this probabilistic sense.  Some more advanced treatment might be found in <a href="http://arxiv.org/abs/1105.0698">Sukhada Fadnavis&#8217;s preprint</a> (which also has something about a &#8220;<a href="http://link.springer.com/chapter/10.1007/3-540-59408-6_66">shameful conjecture</a>&#8221; on chromatic polynomials that was <a href="http://math.nie.edu.sg/fmdong/Research/articles/proof%20of%20a%20chromatic%20polynomial%20conjecture.pdf">proved in 2000</a>, but I don&#8217;t know why it is shameful).   The second part of the talk was on <a href="https://www.sciencemag.org/content/201/4351/131.abstract">problems arising in the study of ESP</a> &#8212; namely that experimental controls are not really present, so the notion of a &#8220;trial&#8221; is hard to pin down, leading (of course) to more false perceptions of coincidences are being surprising.  He closed with some remarks about how our perception of coincidence is really about how our minds work, and pointed to some work by <a href="http://psychology.huji.ac.il/en/?cmd=Faculty.125&amp;act=read&amp;id=75">Ruma Falk</a> for those who are interested in that angle of things.</p>
<p>I was unaware of this body of Diaconis&#8217;s work, and it was nice to have a high-level talk to cap off the day.</p>
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		<title>Capacity-achieving philately</title>
		<link>http://ergodicity.net/2013/05/01/capacity-achieving-philately/</link>
		<comments>http://ergodicity.net/2013/05/01/capacity-achieving-philately/#comments</comments>
		<pubDate>Wed, 01 May 2013 14:44:37 +0000</pubDate>
		<dc:creator>Anand Sarwate</dc:creator>
				<category><![CDATA[Uncategorized]]></category>
		<category><![CDATA[information theory]]></category>
		<category><![CDATA[the post]]></category>

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		<description><![CDATA[The following email came through the ITSOC mailing list, but may be of interest to other readers of the blog. Dear Colleagues, We are making a proposal to the United States Postal Service for the production of a stamp honoring Claude Elwood Shannon on the 100th anniversary of his birth. The proposal is available at: [&#8230;]<img alt="" border="0" src="http://stats.wordpress.com/b.gif?host=ergodicity.net&#038;blog=1934944&#038;post=3690&#038;subd=ergodicwalk&#038;ref=&#038;feed=1" width="1" height="1" />]]></description>
				<content:encoded><![CDATA[<p>The following email came through the ITSOC mailing list, but may be of interest to other readers of the blog.</p>
<blockquote><p>
Dear Colleagues,</p>
<p>We are making a proposal to the United States Postal Service for the production of a stamp honoring Claude Elwood Shannon on the 100th anniversary of his birth.</p>
<p>The proposal is available at: <a href="http://www.itsoc.org/about/shannons-centenary-us-postal-stamp/">http://www.itsoc.org/about/shannons-centenary-us-postal-stamp/</a></p>
<p>Please add your endorsement on that page, and then spread the word!</p>
<p>We would love to have the endorsements of your friends, colleagues, department chair, dean, university president, CEO, government representatives, school-aged children, and the public at large. [Contact information for endorsing individuals will not be posted.]</p>
<p>Thanks for your support!<br />
Best,<br />
Michelle Effros
</p></blockquote>
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		<title>Linkage (desi edition)</title>
		<link>http://ergodicity.net/2013/05/01/linkage-desi-edition/</link>
		<comments>http://ergodicity.net/2013/05/01/linkage-desi-edition/#comments</comments>
		<pubDate>Wed, 01 May 2013 12:42:15 +0000</pubDate>
		<dc:creator>Anand Sarwate</dc:creator>
				<category><![CDATA[Uncategorized]]></category>
		<category><![CDATA[desis]]></category>
		<category><![CDATA[film]]></category>
		<category><![CDATA[news]]></category>
		<category><![CDATA[politics]]></category>

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		<description><![CDATA[An op-ed from n+1 on the safety of being brown. Via Mimosa (I think), a profile of photographer Nemai Ghosh, who worked with Satyajit Ray. Via my father, the story of Indian Jewish actresses in early Bollywood. Things seems to be heating up on the LAC. Not a good sign. The death toll in Dhaka [&#8230;]<img alt="" border="0" src="http://stats.wordpress.com/b.gif?host=ergodicity.net&#038;blog=1934944&#038;post=3658&#038;subd=ergodicwalk&#038;ref=&#038;feed=1" width="1" height="1" />]]></description>
				<content:encoded><![CDATA[<p>An <a href="http://www.nplusonemag.com/white-indians">op-ed from n+1</a> on the safety of being brown.</p>
<p>Via Mimosa (I think), <a href="http://thebigindianpicture.com/2013/01/third-eye/">a profile of photographer Nemai Ghosh</a>, who worked with Satyajit Ray.</p>
<p>Via my father, the story of <a href="http://www.washingtonpost.com/lifestyle/style/shalom-bollywood-reveals-indian-cinemas-surprising-stars-of-its-golden-age/2013/04/18/d043967c-a833-11e2-b029-8fb7e977ef71_story.html">Indian Jewish actresses in early Bollywood</a>.</p>
<p>Things seems to be <a href="https://www.wsws.org/en/articles/2013/04/27/indi-a27.html">heating up on the LAC</a>.  Not a good sign.</p>
<p>The death toll in Dhaka <a href="http://www.bbc.co.uk/news/world-asia-22364891">keeps rising</a>.  This makes <a href="http://www.slate.com/blogs/moneybox/2013/04/24/international_factory_safety.html">Matthew Yglesias&#8217;s reaction</a> (see a stunningly poor example of self-reflection <a href="http://www.slate.com/blogs/moneybox/2013/04/26/some_further_thoughts_on_bangladesh.html">here</a>) a bit more that the <a href="http://inthesetimes.org/duly-noted/entry/14930/no_matt_yglesias_bangladeshi_workers_didnt_choose_to_be_crushed_to_death/">usual</a> <a href="http://crookedtimber.org/2013/04/26/would-it-not-be-easier-for-matt-yglesias-to-dissolve-the-bangladeshi-people-and-elect-another/comment-page-3/">brand</a> of <a href="http://www.mrdestructo.com/2013/04/destructo-salon-does-matthew-yglesias.html">neoliberal odiousness</a>.</p>
<br />Filed under: <a href='http://ergodicity.net/category/uncategorized/'>Uncategorized</a>  <a rel="nofollow" href="http://feeds.wordpress.com/1.0/gocomments/ergodicwalk.wordpress.com/3658/"><img alt="" border="0" src="http://feeds.wordpress.com/1.0/comments/ergodicwalk.wordpress.com/3658/" /></a> <img alt="" border="0" src="http://stats.wordpress.com/b.gif?host=ergodicity.net&#038;blog=1934944&#038;post=3658&#038;subd=ergodicwalk&#038;ref=&#038;feed=1" width="1" height="1" />]]></content:encoded>
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		<title>HGR maximal correlation revisited : a corrected reverse inequality</title>
		<link>http://ergodicity.net/2013/04/27/hgr-maximal-correlation-revisited-a-corrected-reverse-inequality/</link>
		<comments>http://ergodicity.net/2013/04/27/hgr-maximal-correlation-revisited-a-corrected-reverse-inequality/#comments</comments>
		<pubDate>Sun, 28 Apr 2013 00:20:42 +0000</pubDate>
		<dc:creator>Anand Sarwate</dc:creator>
				<category><![CDATA[Uncategorized]]></category>
		<category><![CDATA[information theory]]></category>
		<category><![CDATA[probability]]></category>

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		<description><![CDATA[Sudeep Kamath sent me a note about a recent result he posted on the ArXiV that relates to an earlier post of mine on the HGR maximal correlation and an inequality by Erkip and Cover for Markov chains which I had found interesting: . Since learning about this inequality, I&#8217;ve seen a few talks which [&#8230;]<img alt="" border="0" src="http://stats.wordpress.com/b.gif?host=ergodicity.net&#038;blog=1934944&#038;post=3666&#038;subd=ergodicwalk&#038;ref=&#038;feed=1" width="1" height="1" />]]></description>
				<content:encoded><![CDATA[<p><a href="http://www.eecs.berkeley.edu/~sudeep/">Sudeep Kamath</a> sent me a note about a recent result he posted on the ArXiV that relates to an earlier post of mine on the <a href="http://ergodicity.net/2011/11/02/hgr-maximal-correlation-and-the-ratio-of-mutual-informations/">HGR maximal correlation</a> and an inequality by Erkip and Cover for Markov chains <img src='http://s0.wp.com/latex.php?latex=U+--+X+--+Y&amp;bg=ffffff&amp;fg=29303b&amp;s=0' alt='U -- X -- Y' title='U -- X -- Y' class='latex' /> which I had found interesting:<br />
<img src='http://s0.wp.com/latex.php?latex=I%28U+%3B+Y%29+%5Cle+%5Crho_m%28X%2CY%29%5E2+I%28U+%3B+X%29&amp;bg=ffffff&amp;fg=29303b&amp;s=0' alt='I(U ; Y) &#92;le &#92;rho_m(X,Y)^2 I(U ; X)' title='I(U ; Y) &#92;le &#92;rho_m(X,Y)^2 I(U ; X)' class='latex' />.<br />
Since learning about this inequality, I&#8217;ve seen a few talks which have used the inequality in their proofs, at <a href="http://ergodicity.net/2011/10/14/allerton-2011-2/#more-2506">Allerton in 2011</a> and <a href="http://ergodicity.net/2013/02/18/ita-workshop-2013-post-the-first/">at ITA this year</a>.  Unfortunately, the stated inequality is not correct!</p>
<p><a href="http://arxiv.org/abs/1304.6133">On Maximal Correlation, Hypercontractivity, and the Data Processing Inequality studied by Erkip and Cover</a><br />
Venkat Anantharam, Amin Gohari, Sudeep Kamath, Chandra Nair</p>
<p>What this paper shows is that the inequality is not satisfied with <img src='http://s0.wp.com/latex.php?latex=%5Crho_m%28X%2CY%29%5E2&amp;bg=ffffff&amp;fg=29303b&amp;s=0' alt='&#92;rho_m(X,Y)^2' title='&#92;rho_m(X,Y)^2' class='latex' />, but by another quantity:<br />
<img src='http://s0.wp.com/latex.php?latex=I%28U+%3B+Y%29+%5Cle+s%5E%2A%28X%3BY%29+I%28U+%3B+X%29&amp;bg=ffffff&amp;fg=29303b&amp;s=0' alt='I(U ; Y) &#92;le s^*(X;Y) I(U ; X)' title='I(U ; Y) &#92;le s^*(X;Y) I(U ; X)' class='latex' /><br />
where <img src='http://s0.wp.com/latex.php?latex=s%5E%2A%28X%3BY%29&amp;bg=ffffff&amp;fg=29303b&amp;s=0' alt='s^*(X;Y)' title='s^*(X;Y)' class='latex' /> is given by the following definition.</p>
<blockquote><p>
Let <img src='http://s0.wp.com/latex.php?latex=X&amp;bg=ffffff&amp;fg=29303b&amp;s=0' alt='X' title='X' class='latex' /> and <img src='http://s0.wp.com/latex.php?latex=Y&amp;bg=ffffff&amp;fg=29303b&amp;s=0' alt='Y' title='Y' class='latex' /> be random variables with joint distribution <img src='http://s0.wp.com/latex.php?latex=%28X%2C+Y%29+%5Csim+p%28x%2C+y%29&amp;bg=ffffff&amp;fg=29303b&amp;s=0' alt='(X, Y) &#92;sim p(x, y)' title='(X, Y) &#92;sim p(x, y)' class='latex' />. We define<br />
<img src='http://s0.wp.com/latex.php?latex=s%5E%2A%28X%3BY%29+%3D+%5Csup_%7Br%28x%29+%5Cne+p%28x%29%7D+%5Cfrac%7B+D%28+r%28y%29+%5C%7C+p%28y%29+%29+%7D%7B+D%28+r%28x%29+%5C%7C+p%28x%29+%7D&amp;bg=ffffff&amp;fg=29303b&amp;s=0' alt='s^*(X;Y) = &#92;sup_{r(x) &#92;ne p(x)} &#92;frac{ D( r(y) &#92;| p(y) ) }{ D( r(x) &#92;| p(x) }' title='s^*(X;Y) = &#92;sup_{r(x) &#92;ne p(x)} &#92;frac{ D( r(y) &#92;| p(y) ) }{ D( r(x) &#92;| p(x) }' class='latex' />,<br />
where <img src='http://s0.wp.com/latex.php?latex=r%28y%29&amp;bg=ffffff&amp;fg=29303b&amp;s=0' alt='r(y)' title='r(y)' class='latex' /> denotes the <img src='http://s0.wp.com/latex.php?latex=y&amp;bg=ffffff&amp;fg=29303b&amp;s=0' alt='y' title='y' class='latex' />-marginal distribution of <img src='http://s0.wp.com/latex.php?latex=r%28x%2C+y%29+%3A%3D+r%28x%29p%28y%7Cx%29&amp;bg=ffffff&amp;fg=29303b&amp;s=0' alt='r(x, y) := r(x)p(y|x)' title='r(x, y) := r(x)p(y|x)' class='latex' /> and the supremum on the right hand side is over all probability distributions <img src='http://s0.wp.com/latex.php?latex=r%28x%29&amp;bg=ffffff&amp;fg=29303b&amp;s=0' alt='r(x)' title='r(x)' class='latex' /> that are different from the probability distribution <img src='http://s0.wp.com/latex.php?latex=p%28x%29&amp;bg=ffffff&amp;fg=29303b&amp;s=0' alt='p(x)' title='p(x)' class='latex' />. If either <img src='http://s0.wp.com/latex.php?latex=X&amp;bg=ffffff&amp;fg=29303b&amp;s=0' alt='X' title='X' class='latex' /> or <img src='http://s0.wp.com/latex.php?latex=Y&amp;bg=ffffff&amp;fg=29303b&amp;s=0' alt='Y' title='Y' class='latex' /> is a constant, we define <img src='http://s0.wp.com/latex.php?latex=s%5E%2A%28X%3B+Y%29&amp;bg=ffffff&amp;fg=29303b&amp;s=0' alt='s^*(X; Y)' title='s^*(X; Y)' class='latex' /> to be 0.
</p></blockquote>
<p>Suppose <img src='http://s0.wp.com/latex.php?latex=%28X%2CY%29&amp;bg=ffffff&amp;fg=29303b&amp;s=0' alt='(X,Y)' title='(X,Y)' class='latex' /> have joint distribution <img src='http://s0.wp.com/latex.php?latex=P_%7BXY%7D&amp;bg=ffffff&amp;fg=29303b&amp;s=0' alt='P_{XY}' title='P_{XY}' class='latex' /> (I know I am changing notation here but it&#8217;s easier to explain).  The key to showing their result is through deriving variational characterizations of <img src='http://s0.wp.com/latex.php?latex=%5Crho_m&amp;bg=ffffff&amp;fg=29303b&amp;s=0' alt='&#92;rho_m' title='&#92;rho_m' class='latex' /> and <img src='http://s0.wp.com/latex.php?latex=s%5E%2A&amp;bg=ffffff&amp;fg=29303b&amp;s=0' alt='s^*' title='s^*' class='latex' /> in terms of the function<br />
<img src='http://s0.wp.com/latex.php?latex=t_%7B%5Clambda%7D%28+P_X+%29+%3A%3D+H%28+P_Y+%29+-+%5Clambda+H%28+P_X+%29&amp;bg=ffffff&amp;fg=29303b&amp;s=0' alt='t_{&#92;lambda}( P_X ) := H( P_Y ) - &#92;lambda H( P_X )' title='t_{&#92;lambda}( P_X ) := H( P_Y ) - &#92;lambda H( P_X )' class='latex' /><br />
Rather than getting into that in the blog post, I recommend reading the paper.</p>
<p>The implication of this result is that the inequality of Erkip and Cover is not correct : not only is <img src='http://s0.wp.com/latex.php?latex=%5Crho_m%28X%2CY%29%5E2&amp;bg=ffffff&amp;fg=29303b&amp;s=0' alt='&#92;rho_m(X,Y)^2' title='&#92;rho_m(X,Y)^2' class='latex' /> not the supremum of the ratio, there are distributions for which it is not an upper bound.  The counterexample in the paper is the following: <img src='http://s0.wp.com/latex.php?latex=X+%5Csim+%5Cmathsf%7BBernoulli%7D%281%2F2%29&amp;bg=ffffff&amp;fg=29303b&amp;s=0' alt='X &#92;sim &#92;mathsf{Bernoulli}(1/2)' title='X &#92;sim &#92;mathsf{Bernoulli}(1/2)' class='latex' />, and <img src='http://s0.wp.com/latex.php?latex=Y&amp;bg=ffffff&amp;fg=29303b&amp;s=0' alt='Y' title='Y' class='latex' /> is generated via this asymmetric erasure channel:<br />
<div id="attachment_3670" class="wp-caption aligncenter" style="width: 288px"><a href="http://ergodicwalk.files.wordpress.com/2013/04/erasurehgr.jpg"><img src="http://ergodicwalk.files.wordpress.com/2013/04/erasurehgr.jpg?w=278&#038;h=300" alt="Joint distribution counterexample" width="278" height="300" class="size-medium wp-image-3670" /></a><p class="wp-caption-text">Joint distribution counterexample (Fig. 2 of the paper)</p></div><br />
How can we calculate <img src='http://s0.wp.com/latex.php?latex=%5Crho_m%28X%2CY%29&amp;bg=ffffff&amp;fg=29303b&amp;s=0' alt='&#92;rho_m(X,Y)' title='&#92;rho_m(X,Y)' class='latex' />?  If either <img src='http://s0.wp.com/latex.php?latex=X&amp;bg=ffffff&amp;fg=29303b&amp;s=0' alt='X' title='X' class='latex' /> or <img src='http://s0.wp.com/latex.php?latex=Y&amp;bg=ffffff&amp;fg=29303b&amp;s=0' alt='Y' title='Y' class='latex' /> is binary-valued, then<br />
<img src='http://s0.wp.com/latex.php?latex=%5Crho_m%28X%2CY%29%5E2+%3D+-1+%2B+%5Csum_%7Bx%2Cy%7D+%5Cfrac%7B+p%28x%2Cy%29%5E2+%7D%7B+p%28x%29+p%28y%29+%7D&amp;bg=ffffff&amp;fg=29303b&amp;s=0' alt='&#92;rho_m(X,Y)^2 = -1 + &#92;sum_{x,y} &#92;frac{ p(x,y)^2 }{ p(x) p(y) }' title='&#92;rho_m(X,Y)^2 = -1 + &#92;sum_{x,y} &#92;frac{ p(x,y)^2 }{ p(x) p(y) }' class='latex' /><br />
So for this example <img src='http://s0.wp.com/latex.php?latex=%5Crho_m%28X%2CY%29%5E2+%3D+0.6&amp;bg=ffffff&amp;fg=29303b&amp;s=0' alt='&#92;rho_m(X,Y)^2 = 0.6' title='&#92;rho_m(X,Y)^2 = 0.6' class='latex' />.  However, <img src='http://s0.wp.com/latex.php?latex=s%5E%2A%28+X%2CY%29+%3D+%5Cfrac%7B1%7D%7B2%7D+%5Clog_2%2812%2F5%29+%3E+0.6&amp;bg=ffffff&amp;fg=29303b&amp;s=0' alt='s^*( X,Y) = &#92;frac{1}{2} &#92;log_2(12/5) &gt; 0.6' title='s^*( X,Y) = &#92;frac{1}{2} &#92;log_2(12/5) &gt; 0.6' class='latex' /> and there exists a sequence of variables <img src='http://s0.wp.com/latex.php?latex=U_i&amp;bg=ffffff&amp;fg=29303b&amp;s=0' alt='U_i' title='U_i' class='latex' /> satisfying the Markov chain such that <img src='http://s0.wp.com/latex.php?latex=U_i+--+X+--+Y&amp;bg=ffffff&amp;fg=29303b&amp;s=0' alt='U_i -- X -- Y' title='U_i -- X -- Y' class='latex' /> such that the ratio approaches <img src='http://s0.wp.com/latex.php?latex=s%5E%2A&amp;bg=ffffff&amp;fg=29303b&amp;s=0' alt='s^*' title='s^*' class='latex' />.</p>
<p>So where is the error in the original proof?  Anantharam et al. point to an explanation that the Taylor series expansion used in the proof of the inequality with <img src='http://s0.wp.com/latex.php?latex=%5Crho_m%28X%2CY%29%5E2&amp;bg=ffffff&amp;fg=29303b&amp;s=0' alt='&#92;rho_m(X,Y)^2' title='&#92;rho_m(X,Y)^2' class='latex' /> may not be valid at all points.</p>
<p>This seems to just be the start of a longer story, which I look forward to reading in the future!</p>
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			<media:title type="html">Joint distribution counterexample</media:title>
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		<title>More ArXiV skims</title>
		<link>http://ergodicity.net/2013/04/23/more-arxiv-skims/</link>
		<comments>http://ergodicity.net/2013/04/23/more-arxiv-skims/#comments</comments>
		<pubDate>Tue, 23 Apr 2013 21:37:43 +0000</pubDate>
		<dc:creator>Anand Sarwate</dc:creator>
				<category><![CDATA[Uncategorized]]></category>
		<category><![CDATA[consensus]]></category>
		<category><![CDATA[differential privacy]]></category>
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		<description><![CDATA[Assumptionless consistency of the Lasso Sourav Chatterjee The title says it all. Given -dimensional data points the Lasso tries to fit the model by minimizing the penalized squared error . The paper analyzes the Lasso in the setting where the data are random, so there are i.i.d. copies of a pair of random variables so [&#8230;]<img alt="" border="0" src="http://stats.wordpress.com/b.gif?host=ergodicity.net&#038;blog=1934944&#038;post=3596&#038;subd=ergodicwalk&#038;ref=&#038;feed=1" width="1" height="1" />]]></description>
				<content:encoded><![CDATA[<p><a href="http://arxiv.org/abs/1303.5817">Assumptionless consistency of the Lasso</a><br />
Sourav Chatterjee<br />
The title says it all.  Given <img src='http://s0.wp.com/latex.php?latex=p&amp;bg=ffffff&amp;fg=29303b&amp;s=0' alt='p' title='p' class='latex' />-dimensional data points <img src='http://s0.wp.com/latex.php?latex=%5C%7B+%5Cmathbf%7Bx%7D_i+%3A+i+%5Cin+%5Bn%5D+%5C%7D&amp;bg=ffffff&amp;fg=29303b&amp;s=0' alt='&#92;{ &#92;mathbf{x}_i : i &#92;in [n] &#92;}' title='&#92;{ &#92;mathbf{x}_i : i &#92;in [n] &#92;}' class='latex' /> the Lasso tries to fit the model <img src='http://s0.wp.com/latex.php?latex=%5Cmathbb%7BE%7D%28+y_i+%7C+%5Cmathbf%7Bx_i%7D%29+%3D+%5Cboldsymbol%7B%5Cbeta%7D+%5Cmathbf%7Bx%7D_i&amp;bg=ffffff&amp;fg=29303b&amp;s=0' alt='&#92;mathbb{E}( y_i | &#92;mathbf{x_i}) = &#92;boldsymbol{&#92;beta} &#92;mathbf{x}_i' title='&#92;mathbb{E}( y_i | &#92;mathbf{x_i}) = &#92;boldsymbol{&#92;beta} &#92;mathbf{x}_i' class='latex' /> by minimizing the <img src='http://s0.wp.com/latex.php?latex=%5Cell%5E1&amp;bg=ffffff&amp;fg=29303b&amp;s=0' alt='&#92;ell^1' title='&#92;ell^1' class='latex' /> penalized squared error<br />
<img src='http://s0.wp.com/latex.php?latex=%5Csum_%7Bi%3D1%7D%5E%7Bn%7D+%28y_i+-+%5Cboldsymbol%7B%5Cbeta%7D+%5Cmathbf%7Bx%7D_i%29%5E2+%2B+%5Clambda+%5C%7C+%5Cboldsymbol%7B%5Cbeta%7D+%5C%7C_1&amp;bg=ffffff&amp;fg=29303b&amp;s=0' alt='&#92;sum_{i=1}^{n} (y_i - &#92;boldsymbol{&#92;beta} &#92;mathbf{x}_i)^2 + &#92;lambda &#92;| &#92;boldsymbol{&#92;beta} &#92;|_1' title='&#92;sum_{i=1}^{n} (y_i - &#92;boldsymbol{&#92;beta} &#92;mathbf{x}_i)^2 + &#92;lambda &#92;| &#92;boldsymbol{&#92;beta} &#92;|_1' class='latex' />.<br />
The paper analyzes the Lasso in the setting where the data are random, so there are <img src='http://s0.wp.com/latex.php?latex=n&amp;bg=ffffff&amp;fg=29303b&amp;s=0' alt='n' title='n' class='latex' /> i.i.d. copies of a pair of random variables <img src='http://s0.wp.com/latex.php?latex=%28%5Cmathbf%7BX%7D%2CY%29&amp;bg=ffffff&amp;fg=29303b&amp;s=0' alt='(&#92;mathbf{X},Y)' title='(&#92;mathbf{X},Y)' class='latex' /> so the data is <img src='http://s0.wp.com/latex.php?latex=%5C%7B%28%5Cmathbf%7BX%7D_i%2C+Y_i%29+%3A+i+%5Cin+%5Bn%5D+%5C%7D&amp;bg=ffffff&amp;fg=29303b&amp;s=0' alt='&#92;{(&#92;mathbf{X}_i, Y_i) : i &#92;in [n] &#92;}' title='&#92;{(&#92;mathbf{X}_i, Y_i) : i &#92;in [n] &#92;}' class='latex' />.  The assumptions are on the random variables <img src='http://s0.wp.com/latex.php?latex=%28%5Cmathbf%7BX%7D%2CY%29&amp;bg=ffffff&amp;fg=29303b&amp;s=0' alt='(&#92;mathbf{X},Y)' title='(&#92;mathbf{X},Y)' class='latex' /> : (1) each coordinate <img src='http://s0.wp.com/latex.php?latex=%7CX_i%7C+%5Cle+M&amp;bg=ffffff&amp;fg=29303b&amp;s=0' alt='|X_i| &#92;le M' title='|X_i| &#92;le M' class='latex' /> is bounded, the variable <img src='http://s0.wp.com/latex.php?latex=Y+%3D+%28%5Cboldsymbol%7B%5Cbeta%7D%5E%2A%29%5ET+%5Cmathbf%7BX%7D+%2B+%5Cvarepsilon&amp;bg=ffffff&amp;fg=29303b&amp;s=0' alt='Y = (&#92;boldsymbol{&#92;beta}^*)^T &#92;mathbf{X} + &#92;varepsilon' title='Y = (&#92;boldsymbol{&#92;beta}^*)^T &#92;mathbf{X} + &#92;varepsilon' class='latex' />, and <img src='http://s0.wp.com/latex.php?latex=%5Cvarepsilon+%5Csim+%5Cmathcal%7BN%7D%280%2C%5Csigma%5E2%29&amp;bg=ffffff&amp;fg=29303b&amp;s=0' alt='&#92;varepsilon &#92;sim &#92;mathcal{N}(0,&#92;sigma^2)' title='&#92;varepsilon &#92;sim &#92;mathcal{N}(0,&#92;sigma^2)' class='latex' />, where <img src='http://s0.wp.com/latex.php?latex=%5Cboldsymbol%7B%5Cbeta%7D%5E%2A&amp;bg=ffffff&amp;fg=29303b&amp;s=0' alt='&#92;boldsymbol{&#92;beta}^*' title='&#92;boldsymbol{&#92;beta}^*' class='latex' /> and <img src='http://s0.wp.com/latex.php?latex=%5Csigma&amp;bg=ffffff&amp;fg=29303b&amp;s=0' alt='&#92;sigma' title='&#92;sigma' class='latex' /> are unknown constants.  Basically that&#8217;s all that&#8217;s needed &#8212; given a bound on <img src='http://s0.wp.com/latex.php?latex=%5C%7C%5Cboldsymbol%7B%5Cbeta%7D%5C%7C_1&amp;bg=ffffff&amp;fg=29303b&amp;s=0' alt='&#92;|&#92;boldsymbol{&#92;beta}&#92;|_1' title='&#92;|&#92;boldsymbol{&#92;beta}&#92;|_1' class='latex' />, he derives a bound on the mean-squared prediction error.</p>
<p><a href="http://arxiv.org/abs/1303.5976">On Learnability, Complexity and Stability</a><br />
Silvia Villa, Lorenzo Rosasco, Tomaso Poggio<br />
This is a handy survey on the three topics in the title.  It&#8217;s only 10 pages long, so it&#8217;s a nice fast read.</p>
<p><a href="http://arxiv.org/abs/1303.6149">Adaptivity of averaged stochastic gradient descent to local strong convexity for logistic regression</a><br />
Francis Bach<br />
A central challenge in stochastic optimization is understanding when the convergence rate of the excess loss, which is usually <img src='http://s0.wp.com/latex.php?latex=O%281%2F%5Csqrt%7Bn%7D%29&amp;bg=ffffff&amp;fg=29303b&amp;s=0' alt='O(1/&#92;sqrt{n})' title='O(1/&#92;sqrt{n})' class='latex' />, can be improved to <img src='http://s0.wp.com/latex.php?latex=O%281%2Fn%29&amp;bg=ffffff&amp;fg=29303b&amp;s=0' alt='O(1/n)' title='O(1/n)' class='latex' />.  Most often this involves additional assumptions on the loss functions (which can sometimes get a bit baroque and hard to check).  This paper considers constant step-size algorithms but where instead they consider the averaged iterate $\latex \bar{\theta}_n = \sum_{k=0}^{n-1} \theta_k$.  I&#8217;m trying to slot this in with other things I know about stochastic optimization still, but it&#8217;s definitely worth a skim if you&#8217;re interested in the topic.</p>
<p><a href="http://arxiv.org/abs/1304.2313">On Differentially Private Filtering for Event Streams</a><br />
Jerome Le Ny<br />
Jerome Le Ny has been putting differential privacy into signal processing and control contexts for the past year, and this is another paper in that line of work.  This is important because we&#8217;re still trying to understand how time-series data can be handled in the differential privacy setting.  This paper looks at &#8220;event streams&#8221; which are discrete-valued continuous-time signals (think of count processes), and the problem is to design a differentially private filtering system for such signals.</p>
<p><a href="http://arxiv.org/abs/1304.2268">Gossips and Prejudices: Ergodic Randomized Dynamics in Social Networks</a><br />
Paolo Frasca, Chiara Ravazzi, Roberto Tempo, Hideaki Ishii<br />
This appears to be a gossip version of Acemoglu et al.&#8217;s work on &#8220;stubborn&#8221; agents in the consensus setting.  They show similar qualitative behavior &#8212; opinions fluctuate but their average over time converges (the process is ergodic).  This version of the paper has more of a tutorial feel to it, so the results are a bit easier to parse.</p>
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		<title>Dagadful and nag kesar</title>
		<link>http://ergodicity.net/2013/04/22/dagadful-and-nag-kesar/</link>
		<comments>http://ergodicity.net/2013/04/22/dagadful-and-nag-kesar/#comments</comments>
		<pubDate>Tue, 23 Apr 2013 01:46:18 +0000</pubDate>
		<dc:creator>Anand Sarwate</dc:creator>
				<category><![CDATA[Uncategorized]]></category>
		<category><![CDATA[food]]></category>

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		<description><![CDATA[A few months ago I was home visiting my parents and we had a lunch with a few other Maharashtrians. The conversation turned towards food, and in particular ingredients that are important for making authentic garam masala. Garam masalas vary widely by region in India, and the two ingredients in question were dagadful and nag [&#8230;]<img alt="" border="0" src="http://stats.wordpress.com/b.gif?host=ergodicity.net&#038;blog=1934944&#038;post=3648&#038;subd=ergodicwalk&#038;ref=&#038;feed=1" width="1" height="1" />]]></description>
				<content:encoded><![CDATA[<p>A few months ago I was home visiting my parents and we had a lunch with a few other Maharashtrians.  The conversation turned towards food, and in particular ingredients that are important for making authentic garam masala.  Garam masalas vary widely by region in India, and the two ingredients in question were <i><a href="http://mr.wikipedia.org/wiki/चित्र:Dagadful.JPG">dagadful</a></i> and <i><a href="http://www.nandyala.org/mahanandi/archives/2007/10/14/naaga-keshar-cloves-and-marathi-moggu/">nag kesar</a></i>.  I had never really heard of these spices so I did a bit of research to learn more.</p>
<p>Dagadful (<i>Parmelia perlata</i>) is a lichen, not to be confused with the stone flower <i>Didymocarpus pedicellatus</i>, which is a plant that grows on rocks and is called <i>charela</i> in Hindi, I believe.  The confusing thing is that both plants are used for herbal remedies, but the former is used for culinary purposes.</p>
<p>If you search for &#8220;nag kesar&#8221; you may find <a href="http://en.wikipedia.org/wiki/Mesua_ferrea"><i>Mesua ferrea</a></i>, a hardwood tree that grows in India and surrounds.  That&#8217;s not where the spice comes from, however.  This sparked the most debate at lunch, but I think I&#8217;ve figured out that the <a href="http://www.sciencedirect.com/science/article/pii/S0308814605006692">spice is the bud of a different tree</a>, <i>Mammea longifolia</i>.  Both <i>Mesua</i> and <i><a href="http://en.wikipedia.org/wiki/Mammea">Mammea</a></i> are in the family <a href="http://en.wikipedia.org/wiki/Calophyllaceae"><i>Calophyllaceae</i></a>, which probably led to the name clash.</p>
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		<title>Expected number of faces in Gaussian polytopes</title>
		<link>http://ergodicity.net/2013/04/22/expected-number-of-faces-in-gaussian-polytopes/</link>
		<comments>http://ergodicity.net/2013/04/22/expected-number-of-faces-in-gaussian-polytopes/#comments</comments>
		<pubDate>Mon, 22 Apr 2013 22:02:01 +0000</pubDate>
		<dc:creator>Anand Sarwate</dc:creator>
				<category><![CDATA[Uncategorized]]></category>
		<category><![CDATA[information theory]]></category>
		<category><![CDATA[machine learning]]></category>
		<category><![CDATA[probability]]></category>

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		<description><![CDATA[Last week I was reading Active Learning via Perfect Selective Classification by El-Yaniv and Wiener, and came across a neat result due to Hug and Reitzner that they use in some of their bounds for active learning on Gaussian distributions. The setup is the following : let be jointly Gaussian vectors with distribution in . [&#8230;]<img alt="" border="0" src="http://stats.wordpress.com/b.gif?host=ergodicity.net&#038;blog=1934944&#038;post=3650&#038;subd=ergodicwalk&#038;ref=&#038;feed=1" width="1" height="1" />]]></description>
				<content:encoded><![CDATA[<p>Last week I was reading <a href="http://jmlr.csail.mit.edu/papers/v13/el-yaniv12a.html">Active Learning via Perfect Selective Classification</a> by <a href="http://www.cs.technion.ac.il/~rani/">El-Yaniv</a> and <a href="http://www.informatik.uni-trier.de/~ley/pers/hd/w/Wiener:Yair.html">Wiener</a>, and came across a neat <a href="http://www.jstor.org/stable/30037329">result due to Hug and Reitzner</a> that they use in some of their bounds for active learning on Gaussian distributions.  </p>
<p>The setup is the following : let <img src='http://s0.wp.com/latex.php?latex=X_1%2C+X_2%2C+%5Cldots%2C+X_n&amp;bg=ffffff&amp;fg=29303b&amp;s=0' alt='X_1, X_2, &#92;ldots, X_n' title='X_1, X_2, &#92;ldots, X_n' class='latex' /> be <img src='http://s0.wp.com/latex.php?latex=n&amp;bg=ffffff&amp;fg=29303b&amp;s=0' alt='n' title='n' class='latex' /> jointly Gaussian vectors with distribution <img src='http://s0.wp.com/latex.php?latex=%5Cmathcal%7BN%7D%280%2CI_d%29&amp;bg=ffffff&amp;fg=29303b&amp;s=0' alt='&#92;mathcal{N}(0,I_d)' title='&#92;mathcal{N}(0,I_d)' class='latex' /> in <img src='http://s0.wp.com/latex.php?latex=%5Cmathbb%7BR%7D%5Ed&amp;bg=ffffff&amp;fg=29303b&amp;s=0' alt='&#92;mathbb{R}^d' title='&#92;mathbb{R}^d' class='latex' />.  The convex hull <img src='http://s0.wp.com/latex.php?latex=P_n&amp;bg=ffffff&amp;fg=29303b&amp;s=0' alt='P_n' title='P_n' class='latex' /> of these points is called a <i>Gaussian polytope</i>.  This is a random polytope of course, and we can ask various things about their shape : what is the distribution of the number of vertices, or the number of <img src='http://s0.wp.com/latex.php?latex=k&amp;bg=ffffff&amp;fg=29303b&amp;s=0' alt='k' title='k' class='latex' />-faces?  Let <img src='http://s0.wp.com/latex.php?latex=f_k%28P_n%29&amp;bg=ffffff&amp;fg=29303b&amp;s=0' alt='f_k(P_n)' title='f_k(P_n)' class='latex' /> be the number of <img src='http://s0.wp.com/latex.php?latex=k&amp;bg=ffffff&amp;fg=29303b&amp;s=0' alt='k' title='k' class='latex' />-faces Distributions are hard, but for general <img src='http://s0.wp.com/latex.php?latex=k&amp;bg=ffffff&amp;fg=29303b&amp;s=0' alt='k' title='k' class='latex' /> the expected number of faces (as <img src='http://s0.wp.com/latex.php?latex=n+%5Cto+infty&amp;bg=ffffff&amp;fg=29303b&amp;s=0' alt='n &#92;to infty' title='n &#92;to infty' class='latex' />) is given by</p>
<p><img src='http://s0.wp.com/latex.php?latex=%5Cmathbb%7BE%7D%5B+f_k%28P_n%29%5D+%3D+%5Cfrac%7B2%5Ed%7D%7B%5Csqrt%7Bd%7D%7D+%5Cbinom%7Bd%7D%7Bk%2B1%7D+%5Cbeta_%7Bk%2Cd-1%7D%28%5Cpi+%5Cln+n%29%5E%7B%28d-1%29%2F2%7D+%281+%2B+o%281%29%29&amp;bg=ffffff&amp;fg=29303b&amp;s=0' alt='&#92;mathbb{E}[ f_k(P_n)] = &#92;frac{2^d}{&#92;sqrt{d}} &#92;binom{d}{k+1} &#92;beta_{k,d-1}(&#92;pi &#92;ln n)^{(d-1)/2} (1 + o(1))' title='&#92;mathbb{E}[ f_k(P_n)] = &#92;frac{2^d}{&#92;sqrt{d}} &#92;binom{d}{k+1} &#92;beta_{k,d-1}(&#92;pi &#92;ln n)^{(d-1)/2} (1 + o(1))' class='latex' />,</p>
<p>where <img src='http://s0.wp.com/latex.php?latex=%5Cbeta_%7Bk%2Cd-1%7D&amp;bg=ffffff&amp;fg=29303b&amp;s=0' alt='&#92;beta_{k,d-1}' title='&#92;beta_{k,d-1}' class='latex' /> is the internal angle of a regular <img src='http://s0.wp.com/latex.php?latex=%28d-1%29&amp;bg=ffffff&amp;fg=29303b&amp;s=0' alt='(d-1)' title='(d-1)' class='latex' />-simplex at one of its <img src='http://s0.wp.com/latex.php?latex=k&amp;bg=ffffff&amp;fg=29303b&amp;s=0' alt='k' title='k' class='latex' />-dimensional faces.  What Hug and Reitzner show is a bound on the variance (which then El-Yaniv and Plan use in a Chebyshev bound) : there exists a constant <img src='http://s0.wp.com/latex.php?latex=c_d&amp;bg=ffffff&amp;fg=29303b&amp;s=0' alt='c_d' title='c_d' class='latex' /> such that</p>
<p><img src='http://s0.wp.com/latex.php?latex=%5Cmathrm%7BVar%7D%28+F_k%28P_n%29+%29+%5Cle+c_d+%28%5Cln+n%29%5E%7B%28d-1%29%2F2%7D&amp;bg=ffffff&amp;fg=29303b&amp;s=0' alt='&#92;mathrm{Var}( F_k(P_n) ) &#92;le c_d (&#92;ln n)^{(d-1)/2}' title='&#92;mathrm{Var}( F_k(P_n) ) &#92;le c_d (&#92;ln n)^{(d-1)/2}' class='latex' /></p>
<p>So the variance of the number of <img src='http://s0.wp.com/latex.php?latex=k&amp;bg=ffffff&amp;fg=29303b&amp;s=0' alt='k' title='k' class='latex' />-faces can be upper bounded by something that does not depend at all on the actual value of <img src='http://s0.wp.com/latex.php?latex=k&amp;bg=ffffff&amp;fg=29303b&amp;s=0' alt='k' title='k' class='latex' />.  In fact, they show that </p>
<p><img src='http://s0.wp.com/latex.php?latex=f_k%28P_n%29+%28%5Cln+n%29%5E%7B-%28d-1%29%2F2%7D+%5Cto+%5Cfrac%7B2%5Ed%7D%7B%5Csqrt%7Bd%7D%7D+%5Cbinom%7Bd%7D%7Bk%2B1%7D+%5Cbeta_%7Bk%2Cd-1%7D+%5Cpi%5E%7B%28d-1%29%2F2%7D&amp;bg=ffffff&amp;fg=29303b&amp;s=0' alt='f_k(P_n) (&#92;ln n)^{-(d-1)/2} &#92;to &#92;frac{2^d}{&#92;sqrt{d}} &#92;binom{d}{k+1} &#92;beta_{k,d-1} &#92;pi^{(d-1)/2}' title='f_k(P_n) (&#92;ln n)^{-(d-1)/2} &#92;to &#92;frac{2^d}{&#92;sqrt{d}} &#92;binom{d}{k+1} &#92;beta_{k,d-1} &#92;pi^{(d-1)/2}' class='latex' /></p>
<p>in probability as <img src='http://s0.wp.com/latex.php?latex=n+%5Cto+%5Cinfty&amp;bg=ffffff&amp;fg=29303b&amp;s=0' alt='n &#92;to &#92;infty' title='n &#92;to &#92;infty' class='latex' />.  That is, appropriately normalized, the number of faces converges to a constant.</p>
<p>To me this result was initially surprising, but after some more thought it  makes a bit more sense.  If you give me a cloud of Gaussian points, then I need <img src='http://s0.wp.com/latex.php?latex=k%2B1&amp;bg=ffffff&amp;fg=29303b&amp;s=0' alt='k+1' title='k+1' class='latex' /> points to define a <img src='http://s0.wp.com/latex.php?latex=k&amp;bg=ffffff&amp;fg=29303b&amp;s=0' alt='k' title='k' class='latex' />-face.  The formula for the mean says that if I chose a random set of <img src='http://s0.wp.com/latex.php?latex=k%2B1&amp;bg=ffffff&amp;fg=29303b&amp;s=0' alt='k+1' title='k+1' class='latex' /> points, then approximately <img src='http://s0.wp.com/latex.php?latex=%5Cfrac%7B2%5Ed%7D%7B%5Csqrt%7Bd%7D%7D+%5Cbeta_%7Bk%2Cd-1%7D%28%5Cpi+%5Cln+n%29%5E%7B%28d-1%29%2F2%7D&amp;bg=ffffff&amp;fg=29303b&amp;s=0' alt='&#92;frac{2^d}{&#92;sqrt{d}} &#92;beta_{k,d-1}(&#92;pi &#92;ln n)^{(d-1)/2}' title='&#92;frac{2^d}{&#92;sqrt{d}} &#92;beta_{k,d-1}(&#92;pi &#92;ln n)^{(d-1)/2}' class='latex' /> fraction of them will form a real <img src='http://s0.wp.com/latex.php?latex=k&amp;bg=ffffff&amp;fg=29303b&amp;s=0' alt='k' title='k' class='latex' />-face of the polytope.  This also explains why the simplex-related quantity appears &#8212; points that are on &#8220;opposite sides&#8221; of the sphere (the level sets of the density) are not going to form a face together.  As <img src='http://s0.wp.com/latex.php?latex=n+%5Cto+%5Cinfty&amp;bg=ffffff&amp;fg=29303b&amp;s=0' alt='n &#92;to &#92;infty' title='n &#92;to &#92;infty' class='latex' /> this fraction will change, but effectively the rate of growth in the number of faces with <img src='http://s0.wp.com/latex.php?latex=n&amp;bg=ffffff&amp;fg=29303b&amp;s=0' alt='n' title='n' class='latex' /> is <img src='http://s0.wp.com/latex.php?latex=%28%5Cln+n%29%5E%7B%28d-1%29%2F2%7D&amp;bg=ffffff&amp;fg=29303b&amp;s=0' alt='(&#92;ln n)^{(d-1)/2}' title='(&#92;ln n)^{(d-1)/2}' class='latex' />, regardless of <img src='http://s0.wp.com/latex.php?latex=k&amp;bg=ffffff&amp;fg=29303b&amp;s=0' alt='k' title='k' class='latex' />.</p>
<p>I&#8217;m not sure where this result will be useful for me (yet!) but it seemed like something that the technically-minded readers of the blog would find interesting as well.</p>
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		<title>RAR : a cry of rage</title>
		<link>http://ergodicity.net/2013/04/21/rar-a-cry-of-rage/</link>
		<comments>http://ergodicity.net/2013/04/21/rar-a-cry-of-rage/#comments</comments>
		<pubDate>Sun, 21 Apr 2013 18:14:03 +0000</pubDate>
		<dc:creator>Anand Sarwate</dc:creator>
				<category><![CDATA[Uncategorized]]></category>
		<category><![CDATA[signal processing]]></category>
		<category><![CDATA[information theory]]></category>
		<category><![CDATA[publishing]]></category>
		<category><![CDATA[data compression]]></category>

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		<description><![CDATA[I&#8217;ve been trying to get a camera-ready article for the Signal Processing Magazine and the instructions from IEEE include the following snippet: *VERY IMPORTANT: All source files ( .tex, .doc, .eps, .ps, .bib, .db, .tif, .jpeg, &#8230;) may be uploaded as a single .rar archived file. Please do not attempt to upload files with extensions [&#8230;]<img alt="" border="0" src="http://stats.wordpress.com/b.gif?host=ergodicity.net&#038;blog=1934944&#038;post=3645&#038;subd=ergodicwalk&#038;ref=&#038;feed=1" width="1" height="1" />]]></description>
				<content:encoded><![CDATA[<p>I&#8217;ve been trying to get a camera-ready article for the <a href="http://www.signalprocessingsociety.org/publications/periodicals/spm/">Signal Processing Magazine</a> and the instructions from IEEE include the following snippet:</p>
<blockquote><p>
*VERY IMPORTANT:  All source files ( .tex, .doc, .eps, .ps, .bib, .db, .tif, .jpeg, &#8230;) may be uploaded as a single .rar archived file. Please do not attempt to upload files with extensions .shs, .exe, .com, .vbs, .zip as they are restricted file types.
</p></blockquote>
<p>While I have encountered <code>.rar</code> files before, I was not very familiar with the <a href="http://en.wikipedia.org/wiki/RAR">file format</a> or its history.  I didn&#8217;t know it&#8217;s a proprietary format &#8212; that seems like a weird choice for IEEE to make (although no weirder than PDF perhaps).</p>
<p>What&#8217;s confusing to me is that <a href="http://arxiv.org">ArXiV</a> manages to handle <code>.zip</code> files just fine.  Is <code>.tgz</code> so pass&eacute; now? My experience with RAR is that it is good for compressing (and splitting) large files into easier-to-manage segments.  All of that efficiency seems wasted for a single paper with associated figures and bibliography files and whatnot.</p>
<p>I was trying to find the actual compression algorithm, but like most modern compression software, the innards are a fair bit more complex than the base algorithmic ideas. The Wikipedia article suggests it does a blend of Lempel-Ziv (a variant of LZ77) and prediction by partial matching, but I imagine there&#8217;s a fair bit of tweaking. What I couldn&#8217;t figure out is if there is a new algorithmic idea in there (like in the Burrows-Wheeler Transform (BWT)), or it&#8217;s more a blend of these previous techniques.</p>
<p>Anyway, this silliness means I have to find some extra software to help me compress.  <a href="http://www.macupdate.com/app/mac/21430/simplyrar">SimplyRAR</a> for MacOS seems to work pretty well.</p>
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		<title>Linkage : science edition</title>
		<link>http://ergodicity.net/2013/04/19/linkage-science-edition/</link>
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		<pubDate>Fri, 19 Apr 2013 20:24:27 +0000</pubDate>
		<dc:creator>Anand Sarwate</dc:creator>
				<category><![CDATA[Uncategorized]]></category>
		<category><![CDATA[genomics]]></category>
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		<description><![CDATA[Learning from transcriptomes can be cheaper for organisms which have never been sequenced. A fancy Nature article on mobility privacy, in case you weren&#8217;t convinced by other studies on mobility privacy. Bad statistics in neuroscience. Color me unsurprised. I bet faked results happen a lot in pharmaceutical trials, given the money involved. Perhaps we should [&#8230;]<img alt="" border="0" src="http://stats.wordpress.com/b.gif?host=ergodicity.net&#038;blog=1934944&#038;post=3570&#038;subd=ergodicwalk&#038;ref=&#038;feed=1" width="1" height="1" />]]></description>
				<content:encoded><![CDATA[<p>Learning from transcriptomes can be cheaper <a href="http://www.the-scientist.com/?articles.view/articleNo/34650/title/Sequencing-the-Underdogs/">for organisms which have never been sequenced</a>.</p>
<p>A fancy Nature article on <a href="http://www.nature.com/srep/2013/130325/srep01376/full/srep01376.html">mobility privacy</a>, in case you weren&#8217;t convinced by other studies on mobility privacy.</p>
<p><a href="http://www.the-scientist.com//?articles.view/articleNo/35121/title/Bad-Stats-Plague-Neuroscience/">Bad statistics in neuroscience</a>.  Color me unsurprised.</p>
<p>I bet faked results happen a lot in pharmaceutical trials, given the money involved. Perhaps we should <a href="http://www.the-scientist.com//?articles.view/articleNo/35143/title/Jailed-for-Faking-Data/">jail people for faking data</a> as a disincentive?</p>
<p>The Atheist shoe company <a href="http://www.atheistberlin.com/study">did a study</a> to see if the USPS was discriminating against them.</p>
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