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<!doctype html><html lang="en" class="overflow-y-scroll-ns"><head><meta charSet="utf-8"/><meta name="description" content=" We give a new approach to the dictionary learning (also known as “sparse coding”) problem of recovering an unknown $n\times m$ matrix $A$ (for $m \geq n$) from examples of the form $y = Ax + e,$ where $x$ is a random vector in $\mathbb R^m$ with at most $\tau m$ nonzero coordinates, and $e$ is a random noise vector in $\mathbb R^n$ with bounded magnitude. For the case $m=O(n)$, our algorithm recovers every column of $A$ within arbitrarily good constant accuracy in time $m^{O(\log m/\log(\tau^{-1}))}$, in particular achieving polynomial time if $\tau = m^{-\delta}$ for any $\delta&gt;0$, and time $m^{O(\log m)}$ if $\tau$ is (a sufficiently small) constant. Prior algorithms with comparable assumptions on the distribution required the vector $x$ to be much sparser—at most $\sqrt{n}$ nonzero coordinates—and there were intrinsic barriers preventing these algorithms from applying for denser $x$. We achieve this by designing an algorithm for noisy tensor decomposition that can recover, under quite general conditions, an approximate rank-one decomposition of a tensor $T$, given access to a tensor $T&amp;#39;$ that is $\tau$-close to $T$ in the spectral norm (when considered as a matrix). To our knowledge, this is the first algorithm for tensor decomposition that works in the constant spectral-norm noise regime, where there is no guarantee that the local optima of $T$ and $T&amp;#39;$ have similar structures. Our algorithm is based on a novel approach to using and analyzing the Sum of Squares semidefinite programming hierarchy (Parrilo 2000, Lasserre 2001), and it can be viewed as an indication of the utility of this very general and powerful tool for unsupervised learning problems. "/><meta name="viewport" content="width=device-width, initial-scale=1"/><link rel="icon" href="/images/eth.jpg"/><title>Dictionary learning and tensor decomposition via the sum-of-squares method (David Steurer — ETH Zurich, Computer Science, Theory)</title><style>@font-face{font-family:KaTeX_AMS;src:url(../../katex/dist/fonts/KaTeX_AMS-Regular.woff2) format("woff2"),url(../../katex/dist/fonts/KaTeX_AMS-Regular.woff) format("woff"),url(../../katex/dist/fonts/KaTeX_AMS-Regular.ttf) format("truetype");font-weight:400;font-style:normal;font-display:swap}@font-face{font-family:KaTeX_Caligraphic;src:url(../../katex/dist/fonts/KaTeX_Caligraphic-Bold.woff2) format("woff2"),url(../../katex/dist/fonts/KaTeX_Caligraphic-Bold.woff) format("woff"),url(../../katex/dist/fonts/KaTeX_Caligraphic-Bold.ttf) format("truetype");font-weight:700;font-style:normal;font-display:swap}@font-face{font-family:KaTeX_Caligraphic;src:url(../../katex/dist/fonts/KaTeX_Caligraphic-Regular.woff2) 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STOC 2015.<a href="/paper/soslearning.pdf" class="link b underline hover-dark-blue blue">PDF</a><h2 class="f4 lh-title mid-gray">abstract</h2> We give a new approach to the dictionary learning (also known as “sparse coding”) problem of recovering an unknown <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mo>×</mo><mi>m</mi></mrow><annotation encoding="application/x-tex">n\times m</annotation></semantics></math>n×m matrix <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>A (for <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>m</mi><mo>≥</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">m \geq n</annotation></semantics></math>m≥n) from examples of the form <math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>y</mi><mo>=</mo><mi>A</mi><mi>x</mi><mo>+</mo><mi>e</mi><mo separator="true">,</mo></mrow><annotation encoding="application/x-tex">y = Ax + e,</annotation></semantics></math>y=Ax+e, where <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math>x is a random vector in <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi mathvariant="double-struck">R</mi><mi>m</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb R^m</annotation></semantics></math>Rm with at most <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>τ</mi><mi>m</mi></mrow><annotation encoding="application/x-tex">\tau m</annotation></semantics></math>τm nonzero coordinates, and <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>e</mi></mrow><annotation encoding="application/x-tex">e</annotation></semantics></math>e is a random noise vector in <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi mathvariant="double-struck">R</mi><mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb R^n</annotation></semantics></math>Rn with bounded magnitude. For the case <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>m</mi><mo>=</mo><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">m=O(n)</annotation></semantics></math>m=O(n), our algorithm recovers every column of <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>A within arbitrarily good constant accuracy in time <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>m</mi><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>log</mi><mo>⁡</mo><mi>m</mi><mi mathvariant="normal">/</mi><mi>log</mi><mo>⁡</mo><mo stretchy="false">(</mo><msup><mi>τ</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></msup></mrow><annotation encoding="application/x-tex">m^{O(\log m/\log(\tau^{-1}))}</annotation></semantics></math>mO(logm/log(τ−1)), in particular achieving polynomial time if <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>τ</mi><mo>=</mo><msup><mi>m</mi><mrow><mo>−</mo><mi>δ</mi></mrow></msup></mrow><annotation encoding="application/x-tex">\tau = m^{-\delta}</annotation></semantics></math>τ=m−δ for any <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>δ</mi><mo>></mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\delta>0</annotation></semantics></math>δ>0, and time <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>m</mi><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>log</mi><mo>⁡</mo><mi>m</mi><mo stretchy="false">)</mo></mrow></msup></mrow><annotation encoding="application/x-tex">m^{O(\log m)}</annotation></semantics></math>mO(logm) if <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>τ</mi></mrow><annotation encoding="application/x-tex">\tau</annotation></semantics></math>τ is (a sufficiently small) constant. Prior algorithms with comparable assumptions on the distribution required the vector <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math>x to be much sparser—at most <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msqrt><mi>n</mi></msqrt></mrow><annotation encoding="application/x-tex">\sqrt{n}</annotation></semantics></math>n<svg width='400em' height='1.08em' viewBox='0 0 400000 1080' preserveAspectRatio='xMinYMin slice'><path d='M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z'/></svg> nonzero coordinates—and there were intrinsic barriers preventing these algorithms from applying for denser <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math>x. We achieve this by designing an algorithm for noisy tensor decomposition that can recover, under quite general conditions, an approximate rank-one decomposition of a tensor <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math>T, given access to a tensor <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>T</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup></mrow><annotation encoding="application/x-tex">T'</annotation></semantics></math>T′ that is <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>τ</mi></mrow><annotation encoding="application/x-tex">\tau</annotation></semantics></math>τ-close to <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math>T in the spectral norm (when considered as a matrix). To our knowledge, this is the first algorithm for tensor decomposition that works in the constant spectral-norm noise regime, where there is no guarantee that the local optima of <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math>T and <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>T</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup></mrow><annotation encoding="application/x-tex">T'</annotation></semantics></math>T′ have similar structures. Our algorithm is based on a novel approach to using and analyzing the Sum of Squares semidefinite programming hierarchy (Parrilo 2000, Lasserre 2001), and it can be viewed as an indication of the utility of this very general and powerful tool for unsupervised learning problems. <h2 class="f4 lh-title mid-gray">keywords</h2><ul class="list pa0"><li class="dib mr2 mb2 ba b--mid-gray pa2">sum-of-squares method</li><li class="dib mr2 mb2 ba b--mid-gray pa2">tensor computations</li><li class="dib mr2 mb2 ba b--mid-gray pa2">semidefinite programming</li><li class="dib mr2 mb2 ba b--mid-gray pa2">machine learning</li></ul></article></main><footer><hr class="mv4 bb b--black-10"/><ul class="list db flex justify-center pl0 ma0 lh-copy sans-serif mv4 f5"><li class="dib mr2"><a href="/" class="link underline hover-dark-blue blue">home</a></li><li class="dib mr2"><a href="/papers/" class="link underline hover-dark-blue blue">papers</a></li><li class="dib mr2"><a href="/talks/" class="link underline hover-dark-blue blue">talks</a></li><li class="dib mr2"><a href="/cv/" class="link underline hover-dark-blue blue">c.v.</a></li><li class="dib mr2"><a href="/courses/" class="link underline hover-dark-blue blue">courses</a></li><li class="dib mr2"><a href="/contact/" class="link underline hover-dark-blue blue">contact</a></li><li class="dib mr2"><a href="/events/" class="link underline hover-dark-blue blue">events</a></li></ul></footer></div></body></html>