From 34073a3ef84500623ee742c9e3516f5c1160276f Mon Sep 17 00:00:00 2001 From: Dimitri Lozeve Date: Mon, 30 Jul 2018 17:45:55 +0100 Subject: [PATCH] Dissertation: persistence landscapes --- dissertation/dissertation.pdf | Bin 202558 -> 220895 bytes dissertation/dissertation.tex | 48 ++++++++++++++++++++++++++++++++++ 2 files changed, 48 insertions(+) diff --git a/dissertation/dissertation.pdf b/dissertation/dissertation.pdf index 9a688fe73455b8f9c243664fc48b034624b2187d..a9185f2e0f92cefb7639395d623b76528ed4ec7e 100644 GIT binary patch delta 170718 zcmV(*K;FN;t_?OV%k>oySG=PUFBX89Pt07HOmNd?+%c15@C z;-de*ElKeuQgW^(yD@4ZaOB9cIGi~=b7sWg=I;hKzkXnUynKHAG6HKTAgLtg&F5bY zF`r=b8z@8O)d;1Kwjce|{Bg#Z0>&I;45K-kHz3k`4(*?xzkOh&Rf7>>_D;xu-2O*X zo%-DltQulIz~(0(SOLWL5z`j zD;cd;$)TK>qjteOQI5~v&7+-vKsdvAE7;iwf3V+~rf8=V1Jj4A={Q~Ke4@Nh{Ds$q znzi{?M&E@oSi~5?=B9+3K{(P7prJ=&4j#?%(w+#JjGr*xap|4VRnc2@69kunM_hEN zb#jD9@}Kh(O9QySm)%2!GHxj*V2p21=DQDl2U1X6IU`HBJYx|W&xZVeMEyOwl$mZOdy*xlczvn<;wai%H`Lbh?q;ShJ)s|&>46GF zM3fO@UuXK&Yz?{Q&GhH8npX0xO*!ho7edXK zq)hFF@xc)8A}K$z99-oTI;4FPF87CrM^Ts4kl{ngFn`4KxTlqNKL!C_c|@2Oe<+`0 zZtCE;WnvtEYd)8)a6Rsw?XBUcRAgt)N<{wrTuo{xCV) zECyK#-erBg(Wn0v6lsm5)h4rvyi*OZ7YSZY5MYRtIzMFFzpfJk0waV%&ayuqij9Cl za_~J;!gu=TrXBg_-~L*?5E6gT|(vxXAq<`6vn?L zN*;!P140cA>s1`=(GM>eJie9Gh_+9=M1loXg}s18Cpc-T6IUBilfW{2K-k`)j@9=59g zTW2KlLz@4c8A=JT|%E9o5n-eQg*ARrH!zg%)@+--6i~my;n@o->`f4P_2}pQm+sf_SE1P)w(@ zhA_!$iaOGJ#*3E$j-M&As;Wj~6$uuy@EVHhu-tH8%elQ+-I1IxSPc`R)I+>_tFkrK zY#>}4{$G>7U+wvWewDzowm5dtBD$~iE6+o#)RSg={u0Q5lTI{5JXIO{`2sU4>CN*+mn(~evncJDXozbH2%hG5b-zMf1b2WchgmP z#UIp^dK5_3?LcJQx^DeeK z-r-NEE4YFsf3GX&?~&|6e^;Uq6O{V+2DRMH}5U~`#;#HY8l~A_hX7BwfY%Ao?u51>!@7gKMvP=NJ ziV&~ft*=1$nUu8ix;#1G^$O7rXvkF8lqR3d>}lD;bgOLAZC~%2awk1adC+yW&`mbf zp6)odhlnN?a_O^WJtGyal530528lT(g5M;zWLYz3U9+ok?4MqVoL{c!d_A{iRhW4S zn+e`gZ{%{?R&xMoc>=dQ!+wnsR@;+ieK&vC4Y4_b)H333;s`m*V#yb6qR?TIK{r6b zHccv@>Uq!Sd^11=i`Bb~(#R++If}<0-dD=vE{u^z_{drF`2G7QO>PaioX^OU4*Vv3#vohz~30h*n zuqJM2NlF9(a8A1Ft^*Q+K9;KVPbL!~2N;$OJVO&|>n=Limp`D6c0h4}y7k?Pz zIoMANR6YosEH_6bSExc1;LJX%6V@Wr@Y~4QuE>?M=@X3VTCSX^E5n0_R|Yd7YT>Od zCibdEXyhzL@{>m7z%QIwdbSAas|vHihOV$#QL9yH2e$?njW-wl0>x{r5!$&0)5p++ zp-!v^lf%VRwE%_j4rkNKKY%s@?0=w1=pVGd;vXP6z44?B{^7~`_T|3|TD%s{h4C;P zISa~U@J>XfCJpKK;tRY*3#z2Uu)k8X*Z~?=nc@%%^6la%ODz;Q4zn^)%7|bChw~WSZe__JjkYh?X(yW@>wiIiKaE?gC zkxZML-^d9Hi+Tu#mw1bT{FY_q9)CZN*$mndjjWF)S>Z;!vbzmve|?AenF_m*isI>{C$BZpave|IoMCMn13#ZV}(CRER+`3=m~6< z-6$tKrk$taK0;}vfOt&l!3Mz~BfWW(mH?#4azOR6K3QOKdu}{fC8>7BZ1Y$=$}jPq974*|pkE<>RXV*4MeD8DxCMhX@H@|1F!Mwm9#^}wF2J27 z4?t>Ay%V&z4S&*M&3p?jWT*X?G54w2{?MHzSEg{H#BDXuN(J>H|1d!cuX?dBwRiQ< z-gUbjJAbuf=RWY3jO!QAiW_D9-XHZ-AJSL3*0SWVbl5oAYK{Zxo?Ly|qWrN*-Bw&w zWH(C`g~Kp)U~4AS)kMycJVj^e_0F2<8(4DosG8f4e}9zdlVB3?r+y;S&%Ynqr@y{q zz^SzD2jKYnF}2@W%uaw(Il#qf@8*Riz}yrV=P``#iL#Dae?869%Qsp-zB$>7I7aO? z&b7PR*3C+TgW_-Mp|)DYHmX1> z7pjZ>zM0M3dG+1cJmogDy>C5j2p#Fs)Jwj5az+ksVpne=)7EM34c;q^C%H5cRPMvJ6Sdk2D>~*HLhGC z=cZ{HKH;99DJ|u!5%|zu{SZ4CkV3jmm*kFlpoPM&AJq|VUH84*x)sT~Ki%(4OwOKp zbG0@7`-+B+5NzMTZdvpicFVpA?7h`s=q6t2gX1}2GZWS1d_g(ND1~O;yr~?OhHSQS zRDW1RUltL?5f46feF*K!T+ym_r2&L8cA{f7htN;G%AxznF1@H;JDsEV0qiSHQ=KzD zb?2Sl;rvbcOXFAa&Z1*JXtjm+Z6Se#Kszfr3riKoU&-=p6qxgJ=-nRH zdFF;d|sfACQp7FurRpP(6iH_*LTbEX_N`k5)xEgkJRMcXFlpBq*HZAqVU6D7GQyiUpd7nTO*!`` zHTtO5R^!?k3ETCeosEaAa887%znB#SeAy#JRz%}C%!MHaV+B}F>2inxoN&&JsDGA2 z4Au+l^**X20Xt04F`MO#JQDC)jJs}sEk-o;sSb`QMBvOM5 zy;8ayhO-!%j_X}=I2l)o^hj~#e%vpmQL)CTSdYx3&L%cjR^sBmFJ1gMzfq}9wrXaxu^q-Mt{a!X=E7Bo9sq;0V;(bQ8<0tWG6Hljfs1-6F1wC zl-9dLQU_&KEAwc;pqNj%6KXP!_Uu8RkAJILn}A+h>#wHQsdlHr#F3s!dot9#B?~Jm z0fR!L_f(;zR4^rBc8I+N%>MQLBKDB5lTxmrGbO3hE}u_(SLDDmk!AU zOz6T$9(swStxethH>G>_%Yx`2(U=&%*s9$YEczSIVZK{+*kqpXU@(gszKUr=nuGj$ zd>lL(W04L9=?`;aSI$FU&`9CHjSJQxe_9O!N5?|?OxhRB$58wx&*hBjyf>3|r+qv_ zLn`ueLO!r}a<-sPCDO3evVRbjO^fzXy2l#t)@qj zhi2lS7%&Y3T+94yB6oEBpX7P~d@$|r_7PSIS+vAVthDG~iB!3)-StqMH+r7xa z{+z2T9+f8PIEXG3eqJ<^(XoIV_V?|o zv!2Gd;>eSguS|gwrXX*Xnz2wqJSMf8=>M^5#|NQ(JeFQ|mul5FPxcrps?i6*s6^W7*99p7IPx2ZdOZt8<@D*>sK9n%~@CD^s^; 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100644 --- a/dissertation/dissertation.tex +++ b/dissertation/dissertation.tex @@ -610,21 +610,69 @@ persistence diagrams and Banach spaces. \section{Vectorization methods}% \label{sec:vect-meth} +%% TODO + \subsection{Persistence landscapes} +Persistence landscapes~\cite{bubenik_statistical_2015} are a mean to +project the barcodes in a space where it will be possible to add them +meaningfully. It would thus be possible to define means of persistence +diagrams, along other summary statistics. + +As all the other vectorization techniques mentioned here, this +approach is \emph{injective}, but not surjective, and no explicit +inverse exists to go back from a persistence landscape to the +corresponding persistence diagram. Moreover, a mean of persistence +landscapes do not necessarily have a corresponding persistence +diagram. + +\begin{defn}[Persistence landscape] + The persistence landscape of a diagram $D = \{(b_i,d_i)\}_{i=1}^n$ + is the set of functions $\lambda_k: \mathbb{R} \mapsto \mathbb{R}$, + for $k\in\mathbb{N}$ such that + \[ \lambda_k(x) = k\text{-th largest value of } \{f_{(b_i, + d_i)}(x)\}_{i=1}^n, \] (or zero if the $k$-th largest value does + not exist), where $f_{(b,d)}$ is a piecewise linear function defined by: + \[ f_{(b,d)} = + \begin{cases} + 0& \text{if }x \notin (b,d)\\ + x-b& \text{if }x\in (b,\frac{b+d}{2})\\ + -x+d& \text{if }x\in (\frac{b+d}{2},d). + \end{cases} + \] +\end{defn} + +The persistence landscape is thus a kind of superposition of piecewise +linear functions. Moreover, one can show that persistence landscapes +are stable with respect to the $L^p$ distance, and that the +Wasserstein and bottleneck distances are bounded by the $L^p$ +distance~\cite{bubenik_statistical_2015}. We can thus view the +landscapes as elements of a Banach space in which we can perform the +statistical computations. + \subsection{Persistence images} +\cite{adams_persistence_2017} + \subsection{Tropical and arctic semirings} +\cite{kalisnik_tropical_2018} + \section{Kernel-based methods}% \label{sec:kernel-based-methods} \subsection{Persistent scale-space kernel} +\cite{reininghaus_stable_2015,kwitt_statistical_2015} + \subsection{Persistence weighted gaussian kernel} +\cite{kusano_kernel_2017} + \subsection{Sliced Wasserstein kernel} +\cite{carriere_sliced_2017} + \section{Comparison}% \label{sec:comparison}