Added the Programming assignment from week 4

Stanford_Machine_Learning/Week4/ex3/displayData.m

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+function [h, display_array] = displayData(X, example_width)
+%DISPLAYDATA Display 2D data in a nice grid
+%   [h, display_array] = DISPLAYDATA(X, example_width) displays 2D data
+%   stored in X in a nice grid. It returns the figure handle h and the 
+%   displayed array if requested.
+
+% Set example_width automatically if not passed in
+if ~exist('example_width', 'var') || isempty(example_width) 
+	example_width = round(sqrt(size(X, 2)));
+end
+
+% Gray Image
+colormap(gray);
+
+% Compute rows, cols
+[m n] = size(X);
+example_height = (n / example_width);
+
+% Compute number of items to display
+display_rows = floor(sqrt(m));
+display_cols = ceil(m / display_rows);
+
+% Between images padding
+pad = 1;
+
+% Setup blank display
+display_array = - ones(pad + display_rows * (example_height + pad), ...
+                       pad + display_cols * (example_width + pad));
+
+% Copy each example into a patch on the display array
+curr_ex = 1;
+for j = 1:display_rows
+	for i = 1:display_cols
+		if curr_ex > m, 
+			break; 
+		end
+		% Copy the patch
+
+		% Get the max value of the patch
+		max_val = max(abs(X(curr_ex, :)));
+		display_array(pad + (j - 1) * (example_height + pad) + (1:example_height), ...
+		              pad + (i - 1) * (example_width + pad) + (1:example_width)) = ...
+						reshape(X(curr_ex, :), example_height, example_width) / max_val;
+		curr_ex = curr_ex + 1;
+	end
+	if curr_ex > m, 
+		break; 
+	end
+end
+
+% Display Image
+h = imagesc(display_array, [-1 1]);
+
+% Do not show axis
+axis image off
+
+drawnow;
+
+end

Stanford_Machine_Learning/Week4/ex3/fmincg.m

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+function [X, fX, i] = fmincg(f, X, options, P1, P2, P3, P4, P5)
+% Minimize a continuous differentialble multivariate function. Starting point
+% is given by "X" (D by 1), and the function named in the string "f", must
+% return a function value and a vector of partial derivatives. The Polack-
+% Ribiere flavour of conjugate gradients is used to compute search directions,
+% and a line search using quadratic and cubic polynomial approximations and the
+% Wolfe-Powell stopping criteria is used together with the slope ratio method
+% for guessing initial step sizes. Additionally a bunch of checks are made to
+% make sure that exploration is taking place and that extrapolation will not
+% be unboundedly large. The "length" gives the length of the run: if it is
+% positive, it gives the maximum number of line searches, if negative its
+% absolute gives the maximum allowed number of function evaluations. You can
+% (optionally) give "length" a second component, which will indicate the
+% reduction in function value to be expected in the first line-search (defaults
+% to 1.0). The function returns when either its length is up, or if no further
+% progress can be made (ie, we are at a minimum, or so close that due to
+% numerical problems, we cannot get any closer). If the function terminates
+% within a few iterations, it could be an indication that the function value
+% and derivatives are not consistent (ie, there may be a bug in the
+% implementation of your "f" function). The function returns the found
+% solution "X", a vector of function values "fX" indicating the progress made
+% and "i" the number of iterations (line searches or function evaluations,
+% depending on the sign of "length") used.
+%
+% Usage: [X, fX, i] = fmincg(f, X, options, P1, P2, P3, P4, P5)
+%
+% See also: checkgrad 
+%
+% Copyright (C) 2001 and 2002 by Carl Edward Rasmussen. Date 2002-02-13
+%
+%
+% (C) Copyright 1999, 2000 & 2001, Carl Edward Rasmussen
+% 
+% Permission is granted for anyone to copy, use, or modify these
+% programs and accompanying documents for purposes of research or
+% education, provided this copyright notice is retained, and note is
+% made of any changes that have been made.
+% 
+% These programs and documents are distributed without any warranty,
+% express or implied.  As the programs were written for research
+% purposes only, they have not been tested to the degree that would be
+% advisable in any important application.  All use of these programs is
+% entirely at the user's own risk.
+%
+% [ml-class] Changes Made:
+% 1) Function name and argument specifications
+% 2) Output display
+%
+
+% Read options
+if exist('options', 'var') && ~isempty(options) && isfield(options, 'MaxIter')
+    length = options.MaxIter;
+else
+    length = 100;
+end
+
+
+RHO = 0.01;                            % a bunch of constants for line searches
+SIG = 0.5;       % RHO and SIG are the constants in the Wolfe-Powell conditions
+INT = 0.1;    % don't reevaluate within 0.1 of the limit of the current bracket
+EXT = 3.0;                    % extrapolate maximum 3 times the current bracket
+MAX = 20;                         % max 20 function evaluations per line search
+RATIO = 100;                                      % maximum allowed slope ratio
+
+argstr = ['feval(f, X'];                      % compose string used to call function
+for i = 1:(nargin - 3)
+  argstr = [argstr, ',P', int2str(i)];
+end
+argstr = [argstr, ')'];
+
+if max(size(length)) == 2, red=length(2); length=length(1); else red=1; end
+S=['Iteration '];
+
+i = 0;                                            % zero the run length counter
+ls_failed = 0;                             % no previous line search has failed
+fX = [];
+[f1 df1] = eval(argstr);                      % get function value and gradient
+i = i + (length<0);                                            % count epochs?!
+s = -df1;                                        % search direction is steepest
+d1 = -s'*s;                                                 % this is the slope
+z1 = red/(1-d1);                                  % initial step is red/(|s|+1)
+
+while i < abs(length)                                      % while not finished
+  i = i + (length>0);                                      % count iterations?!
+
+  X0 = X; f0 = f1; df0 = df1;                   % make a copy of current values
+  X = X + z1*s;                                             % begin line search
+  [f2 df2] = eval(argstr);
+  i = i + (length<0);                                          % count epochs?!
+  d2 = df2'*s;
+  f3 = f1; d3 = d1; z3 = -z1;             % initialize point 3 equal to point 1
+  if length>0, M = MAX; else M = min(MAX, -length-i); end
+  success = 0; limit = -1;                     % initialize quanteties
+  while 1
+    while ((f2 > f1+z1*RHO*d1) || (d2 > -SIG*d1)) && (M > 0) 
+      limit = z1;                                         % tighten the bracket
+      if f2 > f1
+        z2 = z3 - (0.5*d3*z3*z3)/(d3*z3+f2-f3);                 % quadratic fit
+      else
+        A = 6*(f2-f3)/z3+3*(d2+d3);                                 % cubic fit
+        B = 3*(f3-f2)-z3*(d3+2*d2);
+        z2 = (sqrt(B*B-A*d2*z3*z3)-B)/A;       % numerical error possible - ok!
+      end
+      if isnan(z2) || isinf(z2)
+        z2 = z3/2;                  % if we had a numerical problem then bisect
+      end
+      z2 = max(min(z2, INT*z3),(1-INT)*z3);  % don't accept too close to limits
+      z1 = z1 + z2;                                           % update the step
+      X = X + z2*s;
+      [f2 df2] = eval(argstr);
+      M = M - 1; i = i + (length<0);                           % count epochs?!
+      d2 = df2'*s;
+      z3 = z3-z2;                    % z3 is now relative to the location of z2
+    end
+    if f2 > f1+z1*RHO*d1 || d2 > -SIG*d1
+      break;                                                % this is a failure
+    elseif d2 > SIG*d1
+      success = 1; break;                                             % success
+    elseif M == 0
+      break;                                                          % failure
+    end
+    A = 6*(f2-f3)/z3+3*(d2+d3);                      % make cubic extrapolation
+    B = 3*(f3-f2)-z3*(d3+2*d2);
+    z2 = -d2*z3*z3/(B+sqrt(B*B-A*d2*z3*z3));        % num. error possible - ok!
+    if ~isreal(z2) || isnan(z2) || isinf(z2) || z2 < 0 % num prob or wrong sign?
+      if limit < -0.5                               % if we have no upper limit
+        z2 = z1 * (EXT-1);                 % the extrapolate the maximum amount
+      else
+        z2 = (limit-z1)/2;                                   % otherwise bisect
+      end
+    elseif (limit > -0.5) && (z2+z1 > limit)         % extraplation beyond max?
+      z2 = (limit-z1)/2;                                               % bisect
+    elseif (limit < -0.5) && (z2+z1 > z1*EXT)       % extrapolation beyond limit
+      z2 = z1*(EXT-1.0);                           % set to extrapolation limit
+    elseif z2 < -z3*INT
+      z2 = -z3*INT;
+    elseif (limit > -0.5) && (z2 < (limit-z1)*(1.0-INT))  % too close to limit?
+      z2 = (limit-z1)*(1.0-INT);
+    end
+    f3 = f2; d3 = d2; z3 = -z2;                  % set point 3 equal to point 2
+    z1 = z1 + z2; X = X + z2*s;                      % update current estimates
+    [f2 df2] = eval(argstr);
+    M = M - 1; i = i + (length<0);                             % count epochs?!
+    d2 = df2'*s;
+  end                                                      % end of line search
+
+  if success                                         % if line search succeeded
+    f1 = f2; fX = [fX' f1]';
+    fprintf('%s %4i | Cost: %4.6e\r', S, i, f1);
+    s = (df2'*df2-df1'*df2)/(df1'*df1)*s - df2;      % Polack-Ribiere direction
+    tmp = df1; df1 = df2; df2 = tmp;                         % swap derivatives
+    d2 = df1'*s;
+    if d2 > 0                                      % new slope must be negative
+      s = -df1;                              % otherwise use steepest direction
+      d2 = -s'*s;    
+    end
+    z1 = z1 * min(RATIO, d1/(d2-realmin));          % slope ratio but max RATIO
+    d1 = d2;
+    ls_failed = 0;                              % this line search did not fail
+  else
+    X = X0; f1 = f0; df1 = df0;  % restore point from before failed line search
+    if ls_failed || i > abs(length)          % line search failed twice in a row
+      break;                             % or we ran out of time, so we give up
+    end
+    tmp = df1; df1 = df2; df2 = tmp;                         % swap derivatives
+    s = -df1;                                                    % try steepest
+    d1 = -s'*s;
+    z1 = 1/(1-d1);                     
+    ls_failed = 1;                                    % this line search failed
+  end
+  if exist('OCTAVE_VERSION')
+    fflush(stdout);
+  end
+end
+fprintf('\n');

Stanford_Machine_Learning/Week4/ex3/lib/jsonlab/AUTHORS.txt

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+The author of "jsonlab" toolbox is Qianqian Fang. Qianqian
+is currently an Assistant Professor at Massachusetts General Hospital, 
+Harvard Medical School.
+
+Address: Martinos Center for Biomedical Imaging, 
+         Massachusetts General Hospital, 
+         Harvard Medical School
+         Bldg 149, 13th St, Charlestown, MA 02129, USA
+URL: http://nmr.mgh.harvard.edu/~fangq/
+Email: <fangq at nmr.mgh.harvard.edu> or <fangqq at gmail.com>
+
+
+The script loadjson.m was built upon previous works by
+
+- Nedialko Krouchev: http://www.mathworks.com/matlabcentral/fileexchange/25713
+       date: 2009/11/02
+- François Glineur: http://www.mathworks.com/matlabcentral/fileexchange/23393
+       date: 2009/03/22
+- Joel Feenstra: http://www.mathworks.com/matlabcentral/fileexchange/20565
+       date: 2008/07/03
+
+
+This toolbox contains patches submitted by the following contributors:
+
+- Blake Johnson <bjohnso at bbn.com>
+  part of revision 341
+
+- Niclas Borlin <Niclas.Borlin at cs.umu.se>
+  various fixes in revision 394, including
+  - loadjson crashes for all-zero sparse matrix.
+  - loadjson crashes for empty sparse matrix.
+  - Non-zero size of 0-by-N and N-by-0 empty matrices is lost after savejson/loadjson.
+  - loadjson crashes for sparse real column vector.
+  - loadjson crashes for sparse complex column vector.
+  - Data is corrupted by savejson for sparse real row vector.
+  - savejson crashes for sparse complex row vector. 
+
+- Yul Kang <yul.kang.on at gmail.com>
+  patches for svn revision 415.
+  - savejson saves an empty cell array as [] instead of null
+  - loadjson differentiates an empty struct from an empty array

Stanford_Machine_Learning/Week4/ex3/lib/jsonlab/ChangeLog.txt

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+============================================================================
+
+   JSONlab - a toolbox to encode/decode JSON/UBJSON files in MATLAB/Octave
+
+----------------------------------------------------------------------------
+
+JSONlab ChangeLog (key features marked by *):
+
+== JSONlab 1.0 (codename: Optimus - Final), FangQ <fangq (at) nmr.mgh.harvard.edu> ==
+
+ 2015/01/02 polish help info for all major functions, update examples, finalize 1.0
+ 2014/12/19 fix a bug to strictly respect NoRowBracket in savejson
+
+== JSONlab 1.0.0-RC2 (codename: Optimus - RC2), FangQ <fangq (at) nmr.mgh.harvard.edu> ==
+
+ 2014/11/22 show progress bar in loadjson ('ShowProgress') 
+ 2014/11/17 add Compact option in savejson to output compact JSON format ('Compact')
+ 2014/11/17 add FastArrayParser in loadjson to specify fast parser applicable levels
+ 2014/09/18 start official github mirror: https://github.com/fangq/jsonlab
+
+== JSONlab 1.0.0-RC1 (codename: Optimus - RC1), FangQ <fangq (at) nmr.mgh.harvard.edu> ==
+
+ 2014/09/17  fix several compatibility issues when running on octave versions 3.2-3.8
+ 2014/09/17  support 2D cell and struct arrays in both savejson and saveubjson
+ 2014/08/04  escape special characters in a JSON string
+ 2014/02/16  fix a bug when saving ubjson files
+
+== JSONlab 0.9.9 (codename: Optimus - beta), FangQ <fangq (at) nmr.mgh.harvard.edu> ==
+
+ 2014/01/22  use binary read and write in saveubjson and loadubjson
+
+== JSONlab 0.9.8-1 (codename: Optimus - alpha update 1), FangQ <fangq (at) nmr.mgh.harvard.edu> ==
+
+ 2013/10/07 better round-trip conservation for empty arrays and structs (patch submitted by Yul Kang)
+
+== JSONlab 0.9.8 (codename: Optimus - alpha), FangQ <fangq (at) nmr.mgh.harvard.edu> ==
+ 2013/08/23 *universal Binary JSON (UBJSON) support, including both saveubjson and loadubjson
+
+== JSONlab 0.9.1 (codename: Rodimus, update 1), FangQ <fangq (at) nmr.mgh.harvard.edu> ==
+ 2012/12/18 *handling of various empty and sparse matrices (fixes submitted by Niclas Borlin)
+
+== JSONlab 0.9.0 (codename: Rodimus), FangQ <fangq (at) nmr.mgh.harvard.edu> ==
+
+ 2012/06/17 *new format for an invalid leading char, unpacking hex code in savejson
+ 2012/06/01  support JSONP in savejson
+ 2012/05/25  fix the empty cell bug (reported by Cyril Davin)
+ 2012/04/05  savejson can save to a file (suggested by Patrick Rapin)
+
+== JSONlab 0.8.1 (codename: Sentiel, Update 1), FangQ <fangq (at) nmr.mgh.harvard.edu> ==
+
+ 2012/02/28  loadjson quotation mark escape bug, see http://bit.ly/yyk1nS
+ 2012/01/25  patch to handle root-less objects, contributed by Blake Johnson
+
+== JSONlab 0.8.0 (codename: Sentiel), FangQ <fangq (at) nmr.mgh.harvard.edu> ==
+
+ 2012/01/13 *speed up loadjson by 20 fold when parsing large data arrays in matlab
+ 2012/01/11  remove row bracket if an array has 1 element, suggested by Mykel Kochenderfer
+ 2011/12/22 *accept sequence of 'param',value input in savejson and loadjson
+ 2011/11/18  fix struct array bug reported by Mykel Kochenderfer
+
+== JSONlab 0.5.1 (codename: Nexus Update 1), FangQ <fangq (at) nmr.mgh.harvard.edu> ==
+
+ 2011/10/21  fix a bug in loadjson, previous code does not use any of the acceleration
+ 2011/10/20  loadjson supports JSON collections - concatenated JSON objects
+
+== JSONlab 0.5.0 (codename: Nexus), FangQ <fangq (at) nmr.mgh.harvard.edu> ==
+
+ 2011/10/16  package and release jsonlab 0.5.0
+ 2011/10/15 *add json demo and regression test, support cpx numbers, fix double quote bug
+ 2011/10/11 *speed up readjson dramatically, interpret _Array* tags, show data in root level
+ 2011/10/10  create jsonlab project, start jsonlab website, add online documentation
+ 2011/10/07 *speed up savejson by 25x using sprintf instead of mat2str, add options support
+ 2011/10/06 *savejson works for structs, cells and arrays
+ 2011/09/09  derive loadjson from JSON parser from MATLAB Central, draft savejson.m

Stanford_Machine_Learning/Week4/ex3/lib/jsonlab/LICENSE_BSD.txt

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+Copyright 2011-2015 Qianqian Fang <fangq at nmr.mgh.harvard.edu>. All rights reserved.
+
+Redistribution and use in source and binary forms, with or without modification, are
+permitted provided that the following conditions are met:
+
+   1. Redistributions of source code must retain the above copyright notice, this list of
+      conditions and the following disclaimer.
+
+   2. Redistributions in binary form must reproduce the above copyright notice, this list
+      of conditions and the following disclaimer in the documentation and/or other materials
+      provided with the distribution.
+
+THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS ''AS IS'' AND ANY EXPRESS OR IMPLIED
+WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND
+FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDERS 
+OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR
+CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR
+SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON
+ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING
+NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF
+ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
+
+The views and conclusions contained in the software and documentation are those of the
+authors and should not be interpreted as representing official policies, either expressed
+or implied, of the copyright holders.