forked from lakshmiDRIP/DROP
-
Notifications
You must be signed in to change notification settings - Fork 0
/
Copy pathStatisticalLearningModule.html
839 lines (672 loc) · 38 KB
/
StatisticalLearningModule.html
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
827
828
829
830
831
832
833
834
835
836
837
838
839
<!doctype html>
<html>
<head>
<meta charset="utf-8">
<meta http-equiv="X-UA-Compatible" content="chrome=1">
<title>DROP Statistical Learning</title>
<link rel="stylesheet" href="stylesheets/styles.css">
<link rel="stylesheet" href="stylesheets/github-dark.css">
<script src="javascripts/scale.fix.js"></script>
<meta name="viewport" content="width=device-width, initial-scale=1, user-scalable=no">
<!--[if lt IE 9]>
<script src="//html5shiv.googlecode.com/svn/trunk/html5.js"></script>
<![endif]-->
</head>
<body>
<div class="wrapper">
<header>
<h1>DROP Statistical Learning</h1>
<p>DROP Statistical Learning</p>
<p class="view"><a href="https://github.com/lakshmiDRIP/DRIP-Statistical-Learning">View the Project on GitHub <small>lakshmiDRIP/DRIP-Statistical-Learning</small></a></p>
<ul>
<li><a href="https://github.com/lakshmiDRIP/DROP/zipball/master">Download <strong>ZIP File</strong></a></li>
<li><a href="https://github.com/lakshmiDRIP/DROP/tarball/master">Download <strong>TAR Ball</strong></a></li>
<li><a href="https://github.com/lakshmiDRIP/DROP">View On <strong>GitHub</strong></a></li>
</ul>
</header>
<section>
<p align="center"><img src="https://github.com/lakshmiDRIP/DROP/blob/master/DRIP_Logo.gif?raw=true" width="100"></p>
<p><strong>v4.45</strong> <em>14 November 2019</em></p>
<p>DROP Statistical Learning is a collection of Java libraries for Statistical Evaluation and Machine Learning.</p>
<p>DROP Statistical Learning is composed of the following main libraries:</p>
<ul>
<li>Probabilistic Sequence Measure Concentration Bounds Library</li>
<li>Statistical Learning Theory Framework Library</li>
<li>Empirical Risk Minimization Library</li>
<li>VC and Capacity Measure Library</li>
<li>Covering Numbers Library</li>
<li>Alternate Statistical Learning Library</li>
<li>Problem Space and Algorithms Families Library</li>
<li>Parametric Classification Library</li>
<li>Non-parametric Classification Library</li>
<li>Clustering Library</li>
<li>Ensemble Learning Library</li>
<li>Multi-linear Sub-space Learning Library</li>
<li>Real-Valued Sequence Learning Library</li>
<li>Real-Valued Learning Library</li>
<li>Sequence Labeling Library</li>
<li>Bayesian Library</li>
<li>Linear Algebra Support Library</li>
</ul>
<p>For Installation, Documentation and Samples, and the associated supporting Numerical Libraries please check out <a href="https://github.com/lakshmiDRIP/DROP">DROP</a>.</p>
<h2>
<a id="drip-core-technical-specifications" class="anchor" href="#drip-core-technical-specifications" aria-hidden="true"><span aria-hidden="true" class="octicon octicon-link"></span></a>DROP Core Technical Specifications</h2>
<ul>
<li><a href="https://github.com/lakshmiDRIP/DROP/tree/master/Docs/Internal/AssetAllocation/AssetAllocation_v3.08.pdf">Asset Allocation Library</a></li>
<li><a href="https://github.com/lakshmiDRIP/DROP/tree/master/Docs/Internal/FixedIncome/FixedIncomeAnalytics_v3.11.pdf">Fixed Income Analytics</a></li>
<li><a href="https://github.com/lakshmiDRIP/DROP/tree/master/Docs/Internal/TransactionCost/TransactionCostAnalytics_v3.10.pdf">Transaction Cost Analytics</a></li>
<li><a href="https://github.com/lakshmiDRIP/DROP/tree/master/Docs/Internal/XVA/XVAAnalytics_v3.87.pdf">XVA Analytics</a></li>
<li><a href="https://github.com/lakshmiDRIP/DROP/tree/master/Docs/Internal/Exposure/ExposureAnalytics_v4.01.pdf">Exposure Analytics</a></li>
</ul>
<h2>
<a id="drip-supporting-technical-specifications" class="anchor" href="#drip-supporting-technical-specifications" aria-hidden="true"><span aria-hidden="true" class="octicon octicon-link"></span></a>DROP Supporting Technical Specifications</h2>
<ul>
<li><a href="https://github.com/lakshmiDRIP/DROP/tree/master/Docs/Internal/ModelValidation/ModelValidationAnalytics_v4.12.pdf">Model Validation Library</a></li>
<li><a href="https://github.com/lakshmiDRIP/DROP/tree/master/Docs/Internal/SplineBuilder/SplineBuilder_v3.14.pdf">Spline Builder Library</a></li>
<li><a href="https://github.com/lakshmiDRIP/DROP/tree/master/Docs/Internal/NumericalOptimizer/NumericalOptimization_v4.28.pdf">Numerical Optimization Library</a></li>
<li><a href="https://github.com/lakshmiDRIP/DROP/tree/master/Docs/Internal/StatisticalLearning/StatisticalLearningLibrary_v3.93.pdf">Statistical Learning Library</a></li>
<li><a href="https://github.com/lakshmiDRIP/DROP/tree/master/Docs/Internal/MachineLearning/MachineLearningLibrary_v3.85.pdf">Machine Learning Library</a></li>
</ul>
<h2>
<a id="additional-documentation" class="anchor" href="#additional-documentation" aria-hidden="true"><span aria-hidden="true" class="octicon octicon-link"></span></a>Additional Documentation</h2>
<ul>
<li><a href="https://github.com/lakshmiDRIP/DROP">DROP GitHub Source</a></li>
<li><a href="https://lakshmidrip.github.io/DROP/Javadoc/index.html">DROP API Javadoc</a></li>
<li><a href="https://github.com/lakshmiDRIP/DROP/tree/master/ReleaseNotes">DROP Release Notes</a></li>
<li><a href="https://github.com/lakshmiDRIP/DROP/tree/master/Docs/Internal">DROP Technical Specifications</a></li>
<li><a href="https://github.com/lakshmiDRIP/DROP/tree/master/Docs/External">DROP External Specifications</a></li>
<li>User guide is a work in progress!</li>
</ul>
<h2>
<a id="samples-statistical-learning---need-much-much-more" class="anchor" href="#samples-statistical-learning---need-much-much-more" aria-hidden="true"><span aria-hidden="true" class="octicon octicon-link"></span></a>Samples (Statistical Learning - Need much, much more!)</h2>
<ul>
<li><a href="https://github.com/lakshmiDRIP/DROP/tree/master/src/main/java/org/drip/sample/efronstein">Efron Stein Bounds</a></li>
<li><a href="https://github.com/lakshmiDRIP/DROP/tree/master/src/main/java/org/drip/sample/sequence">Custom Sequence Bounds</a></li>
<li><a href="https://github.com/lakshmiDRIP/DROP/tree/master/src/main/java/org/drip/sample/coveringnumber">Covering Number Bounds</a></li>
<li><a href="https://github.com/lakshmiDRIP/DROP/tree/master/src/main/java/org/drip/sample/classifier">Binary Classifier Bounds</a></li>
</ul>
<h2>
<a id="samples-numerical-support---need-much-more" class="anchor" href="#samples-numerical-support---need-much-more" aria-hidden="true"><span aria-hidden="true" class="octicon octicon-link"></span></a>Samples (Numerical Support - Need much more!)</h2>
<ul>
<li><a href="https://github.com/lakshmiDRIP/DROP/tree/master/src/main/java/org/drip/sample/matrix">Linear Algebra/Components</a></li>
<li><a href="https://github.com/lakshmiDRIP/DROP/tree/master/src/main/java/org/drip/sample/measure">Closed Distribution Measure</a></li>
<li><a href="https://github.com/lakshmiDRIP/DROP/tree/master/src/main/java/org/drip/sample/statistics">Empirical Distribution Measure</a></li>
<li><a href="https://github.com/lakshmiDRIP/DROP/tree/master/src/main/java/org/drip/sample/numerical">Search/Quadrature/Fourier</a></li>
<li><a href="https://github.com/lakshmiDRIP/DROP/tree/master/src/main/java/org/drip/sample/numeraire">R<sup>x</sup> Diffusion/Jump/Joint Numeraire</a></li>
<li><a href="https://github.com/lakshmiDRIP/DROP/tree/master/src/main/java/org/drip/sample/rng">Quadratic Re-sampling Unbiased Antithetic Random Generator</a></li>
</ul>
<h2>
<a id="samples-misc---need-much-more" class="anchor" href="#samples-misc---need-much-more" aria-hidden="true"><span aria-hidden="true" class="octicon octicon-link"></span></a>Samples (Miscellaneous - Need much more!)</h2>
<ul>
<li><a href="https://github.com/lakshmiDRIP/DROP/tree/master/src/main/java/org/drip/sample/algo">C<sup>x</sup> R<sup>x</sup> In-Place Manipulation</a></li>
<li><a href="https://github.com/lakshmiDRIP/DROP/tree/master/src/main/java/org/drip/sample/date">Calendar Date Roll Day Count</a></li>
<li><a href="https://github.com/lakshmiDRIP/DROP/tree/master/src/main/java/org/drip/sample/env">Environment Module Loader Cache Manager</a></li>
<li><a href="https://github.com/lakshmiDRIP/DROP/tree/master/src/main/java/org/drip/sample/json">RFC4627 Compliant JSON Lexer Serializer</a></li>
<li><a href="https://github.com/lakshmiDRIP/DROP/tree/master/src/main/java/org/drip/sample/service">Curve Product Portfolio Valuation Service Templates</a></li>
<li><a href="https://github.com/lakshmiDRIP/DROP/tree/master/src/main/java/org/drip/sample/graph">Shortest Path Graph Navigation Algorithms</a></li>
</ul>
<h2>
<a id="features" class="anchor" href="#features" aria-hidden="true"><span aria-hidden="true" class="octicon octicon-link"></span></a>Features</h2>
<h3>
<a id="probabilistic-bounds-and-concentration-of-measure-sequences" class="anchor" href="#probabilistic-bounds-and-concentration-of-measure-sequences" aria-hidden="true"><span aria-hidden="true" class="octicon octicon-link"></span></a>Probabilistic Bounds and Concentration of Measure Sequences</h3>
<h4>
<a id="probabilistic-bounds" class="anchor" href="#probabilistic-bounds" aria-hidden="true"><span aria-hidden="true" class="octicon octicon-link"></span></a>Probabilistic Bounds</h4>
<ul>
<li>Tail Probability Bounds Estimation</li>
<li>Basic Probability Inequalities</li>
<li>Cauchy-Schwartz Inequality</li>
<li>Association Inequalities</li>
<li>Moment, Gaussian, and Exponential Bounds</li>
<li>Bounding Sums of Independent Random Variables</li>
<li>Non Moment Based Bounding - Hoeffding Bound</li>
<li>Moment Based Bounds</li>
<li>Binomial Tails</li>
<li>Custom Bounds for Special i.i.d. Sequences</li>
</ul>
<h4>
<a id="efron-stein-bounds" class="anchor" href="#efron-stein-bounds" aria-hidden="true"><span aria-hidden="true" class="octicon octicon-link"></span></a>Efron Stein Bounds</h4>
<ul>
<li>Martingale Differences Sum Inequality</li>
<li>Efron-Stein Inequality</li>
<li>Bounded Differences Inequality</li>
<li>Bounded Differences Inequality - Applications</li>
<li>Self-Bounding Functions</li>
<li>Configuration Functions</li>
</ul>
<h4>
<a id="entropy-methods" class="anchor" href="#entropy-methods" aria-hidden="true"><span aria-hidden="true" class="octicon octicon-link"></span></a>Entropy Methods</h4>
<ul>
<li>Information Theory - Basics</li>
<li>Tensorization of the Entropy</li>
<li>Logarithmic Sobolev Inequalities</li>
<li>Logarithmic Sobolev Inequalities - Applications</li>
<li>Exponential Inequalities for Self-Bounding Functions</li>
<li>Combinatorial Entropy</li>
<li>Variations on the Theme of Self-Bounding Functions</li>
</ul>
<h4>
<a id="concentration-of-measure" class="anchor" href="#concentration-of-measure" aria-hidden="true"><span aria-hidden="true" class="octicon octicon-link"></span></a>Concentration of Measure</h4>
<ul>
<li>Equivalent Bounded Differences Inequality</li>
<li>Convex Distance Inequality</li>
<li>Convex Distance Inequality - Proof</li>
<li>Application of the Convex Distance Inequality - Bin Packing</li>
</ul>
<h3>
<a id="statistical-learning-theory---foundation-and-framework" class="anchor" href="#statistical-learning-theory---foundation-and-framework" aria-hidden="true"><span aria-hidden="true" class="octicon octicon-link"></span></a>Statistical Learning Theory - Foundation and Framework</h3>
<h4>
<a id="standard-slt-framework" class="anchor" href="#standard-slt-framework" aria-hidden="true"><span aria-hidden="true" class="octicon octicon-link"></span></a>Standard SLT Framework</h4>
<ul>
<li>Computational Learning Theory</li>
<li>Probably Approximately Correct (PAC) Learning</li>
<li>PAC Definitions and Terminology</li>
<li>SLT Setup</li>
<li>Algorithms for Reducing Over-fitting</li>
<li>Bayesian Normalized Regularizer Setup</li>
</ul>
<h4>
<a id="generalization-and-consistency" class="anchor" href="#generalization-and-consistency" aria-hidden="true"><span aria-hidden="true" class="octicon octicon-link"></span></a>Generalization and Consistency</h4>
<ul>
<li>Types of Consistency</li>
<li>Bias-Variance or Estimation-Approximation Trade-off</li>
<li>Bias-Variance Decomposition</li>
<li>Bias-Variance Optimization</li>
<li>Generalization and Consistency for kNN</li>
</ul>
<h3>
<a id="empirical-risk-minimization---principles-and-techniques" class="anchor" href="#empirical-risk-minimization---principles-and-techniques" aria-hidden="true"><span aria-hidden="true" class="octicon octicon-link"></span></a>Empirical Risk Minimization - Principles and Techniques</h3>
<h4>
<a id="empirical-risk-minimization" class="anchor" href="#empirical-risk-minimization" aria-hidden="true"><span aria-hidden="true" class="octicon octicon-link"></span></a>Empirical Risk Minimization</h4>
<ul>
<li>Overview</li>
<li>The Loss Functions and Empirical Risk Minimization Principles</li>
<li>Application of the Central Limit Theorem (CLT) and the Law of Large Numbers (LLN)</li>
<li>Inconsistency of Empirical Risk Minimizers</li>
<li>Uniform Convergence</li>
<li>ERM Complexity</li>
</ul>
<h4>
<a id="symmetrization" class="anchor" href="#symmetrization" aria-hidden="true"><span aria-hidden="true" class="octicon octicon-link"></span></a>Symmetrization</h4>
<ul>
<li>The Symmetrization Lemma</li>
</ul>
<h4>
<a id="generalization-bounds" class="anchor" href="#generalization-bounds" aria-hidden="true"><span aria-hidden="true" class="octicon octicon-link"></span></a>Generalization Bounds</h4>
<ul>
<li>The Union Bound</li>
<li>Shattering Coefficient</li>
<li>Empirical Risk Generalization Bound</li>
<li>Large Margin Bounds</li>
</ul>
<h4>
<a id="rademacher-complexity" class="anchor" href="#rademacher-complexity" aria-hidden="true"><span aria-hidden="true" class="octicon octicon-link"></span></a>Rademacher Complexity</h4>
<ul>
<li>Rademacher-based Uniform Convergence</li>
<li>VC Entropy</li>
<li>Chaining Technique</li>
</ul>
<h4>
<a id="local-rademacher-averages" class="anchor" href="#local-rademacher-averages" aria-hidden="true"><span aria-hidden="true" class="octicon octicon-link"></span></a>Local Rademacher Averages</h4>
<ul>
<li>Star-Hull and Sub-Root Functions</li>
<li>Local Rademacher Averages and Fixed Point</li>
<li>Local Rademacher Averages - Consequences</li>
</ul>
<h4>
<a id="normalized-erm" class="anchor" href="#normalized-erm" aria-hidden="true"><span aria-hidden="true" class="octicon octicon-link"></span></a>Normalized ERM</h4>
<ul>
<li>Computing the Normalized Empirical Risk Bounds</li>
<li>De-normalized Bounds</li>
</ul>
<h4>
<a id="noise-conditions" class="anchor" href="#noise-conditions" aria-hidden="true"><span aria-hidden="true" class="octicon octicon-link"></span></a>Noise Conditions</h4>
<ul>
<li>SLT Analysis Metrics</li>
<li>Types of Noise Conditions</li>
<li>Relative Loss Class</li>
</ul>
<h3>
<a id="vc-theory-and-capacity-measure-analysis" class="anchor" href="#vc-theory-and-capacity-measure-analysis" aria-hidden="true"><span aria-hidden="true" class="octicon octicon-link"></span></a>VC Theory and Capacity Measure Analysis</h3>
<h4>
<a id="vc-theory-and-vc-dimension" class="anchor" href="#vc-theory-and-vc-dimension" aria-hidden="true"><span aria-hidden="true" class="octicon octicon-link"></span></a>VC Theory and VC Dimension</h4>
<ul>
<li>Empirical Processes</li>
<li>Bounding the Empirical Loss Function</li>
<li>VC Dimension - Setup</li>
<li>Incorporating the Formal VC Definition</li>
<li>VC Dimension Examples</li>
<li>VC Dimension vs. Popper's Dimension</li>
</ul>
<h4>
<a id="sauer-lemma-and-vc-classifier-framework" class="anchor" href="#sauer-lemma-and-vc-classifier-framework" aria-hidden="true"><span aria-hidden="true" class="octicon octicon-link"></span></a>Sauer Lemma and VC Classifier Framework</h4>
<ul>
<li>Working out Sauer Lemma Bounds</li>
<li>Sauer Lemma ERM Bounds</li>
<li>VC Index</li>
<li>VC Classifier Framework</li>
</ul>
<h3>
<a id="capacitycomplexity-estimation-using-covering-numbers" class="anchor" href="#capacitycomplexity-estimation-using-covering-numbers" aria-hidden="true"><span aria-hidden="true" class="octicon octicon-link"></span></a>Capacity/Complexity Estimation Using Covering Numbers</h3>
<h4>
<a id="covering-and-entropy-numbers" class="anchor" href="#covering-and-entropy-numbers" aria-hidden="true"><span aria-hidden="true" class="octicon octicon-link"></span></a>Covering and Entropy Numbers</h4>
<ul>
<li>Nomenclature- Normed Spaces</li>
<li>Covering, Entropy, and Dyadic Numbers</li>
<li>Background and Overview of Basic Results</li>
</ul>
<h4>
<a id="covering-numbers-for-real-valued-function-classes" class="anchor" href="#covering-numbers-for-real-valued-function-classes" aria-hidden="true"><span aria-hidden="true" class="octicon octicon-link"></span></a>Covering Numbers for Real-Valued Function Classes</h4>
<ul>
<li>Functions of Bounded Variation</li>
<li>Functions of Bounded Variation - Upper Bound</li>
<li>Functions of Bounded Variation - Lower Bound</li>
<li>General Function Classes</li>
<li>General Function Class Bounds</li>
<li>General Function Class Bounds - Lemmas</li>
<li>General Function Class - Upper Bounds</li>
<li>General Function Class - Lower Bounds</li>
</ul>
<h4>
<a id="operator-theory-methods-for-entropy-numbers" class="anchor" href="#operator-theory-methods-for-entropy-numbers" aria-hidden="true"><span aria-hidden="true" class="octicon octicon-link"></span></a>Operator Theory Methods for Entropy Numbers</h4>
<ul>
<li>Generalization Bounds via Uniform Convergence</li>
<li>Basic Uniform Convergence Bounds</li>
<li>Loss Function Induced Classes</li>
<li>Standard Form of Uniform Convergence</li>
</ul>
<h4>
<a id="kernel-machines" class="anchor" href="#kernel-machines" aria-hidden="true"><span aria-hidden="true" class="octicon octicon-link"></span></a>Kernel Machines</h4>
<ul>
<li>SVM Capacity Control</li>
<li>Nonlinear Kernels</li>
<li>Generalization Performance of Regularization Networks</li>
<li>Covering Number Determination Steps</li>
<li>Challenges Presenting Master Generalization Error</li>
</ul>
<h4>
<a id="entropy-number-for-kernel-machines" class="anchor" href="#entropy-number-for-kernel-machines" aria-hidden="true"><span aria-hidden="true" class="octicon octicon-link"></span></a>Entropy Number for Kernel Machines</h4>
<ul>
<li>Mercer Kernels</li>
<li>Equivalent Kernels</li>
<li>Mapping Phi into L2</li>
<li>Corrigenda to the Mercer Conditions</li>
<li>L2 Unit Ball -> Epsilon Mapping Scaling Operator</li>
<li>Unit Bounding Operator Entropy Numbers</li>
<li>The SVM Operator</li>
<li>Maurey's Theorem</li>
<li>Bounds for SV Classes</li>
<li>Asymptotic Rates of Decay for the Entropy Numbers</li>
</ul>
<h4>
<a id="discrete-spectra-of-convolution-operators" class="anchor" href="#discrete-spectra-of-convolution-operators" aria-hidden="true"><span aria-hidden="true" class="octicon octicon-link"></span></a>Discrete Spectra of Convolution Operators</h4>
<ul>
<li>Kernels with Compact/Non-compact Support</li>
<li>The Kernel Operator Eigenvalues</li>
<li>Choosing Nu</li>
<li>Extensions to d-dimensions</li>
</ul>
<h4>
<a id="covering-numbers-for-given-decay-rates" class="anchor" href="#covering-numbers-for-given-decay-rates" aria-hidden="true"><span aria-hidden="true" class="octicon octicon-link"></span></a>Covering Numbers for Given Decay Rates</h4>
<ul>
<li>Asymptotic/Non-asymptotic Decay of Covering Numbers</li>
<li>Polynomial Eigenvalue Decay</li>
<li>Summation and Integration of Non-decreasing Functions</li>
<li>Exponential Polynomial Decay</li>
</ul>
<h4>
<a id="kernels-for-high-dimensional-data" class="anchor" href="#kernels-for-high-dimensional-data" aria-hidden="true"><span aria-hidden="true" class="octicon octicon-link"></span></a>Kernels for High-Dimensional Data</h4>
<ul>
<li>Kernel Fourier Transforms</li>
<li>Degenerate Kernel Bounds</li>
<li>Covering Numbers for Degenerate Systems</li>
<li>Bounds for Kernels in R^d</li>
<li>Impact of the Fourier Transform Decay on the Entropy Numbers</li>
</ul>
<h4>
<a id="regularization-networks-entropy-numbers-determination---practice" class="anchor" href="#regularization-networks-entropy-numbers-determination---practice" aria-hidden="true"><span aria-hidden="true" class="octicon octicon-link"></span></a>Regularization Networks Entropy Numbers Determination - Practice</h4>
<ul>
<li>Custom Applications of the Kernel Machines Entropy Numbers</li>
<li>Extensions to the Operator-Theoretic Viewpoint for Covering Numbers</li>
</ul>
<h3>
<a id="alternate-statistical-learning-approaches" class="anchor" href="#alternate-statistical-learning-approaches" aria-hidden="true"><span aria-hidden="true" class="octicon octicon-link"></span></a>Alternate Statistical Learning Approaches</h3>
<h4>
<a id="minimum-description-length-approach" class="anchor" href="#minimum-description-length-approach" aria-hidden="true"><span aria-hidden="true" class="octicon octicon-link"></span></a>Minimum Description Length Approach</h4>
<ul>
<li>Coding Approaches</li>
<li>MDL Analyses</li>
</ul>
<h4>
<a id="bayesian-methods" class="anchor" href="#bayesian-methods" aria-hidden="true"><span aria-hidden="true" class="octicon octicon-link"></span></a>Bayesian Methods</h4>
<ul>
<li>Bayesian and Frequentist Approaches</li>
<li>Bayesian Approaches</li>
</ul>
<h4>
<a id="knowledge-based-bounds" class="anchor" href="#knowledge-based-bounds" aria-hidden="true"><span aria-hidden="true" class="octicon octicon-link"></span></a>Knowledge Based Bounds</h4>
<ul>
<li>Places to Incorporate Bounds</li>
<li>Prior Knowledge into the Function Space</li>
</ul>
<h4>
<a id="approximation-error-and-bayes-consistency" class="anchor" href="#approximation-error-and-bayes-consistency" aria-hidden="true"><span aria-hidden="true" class="octicon octicon-link"></span></a>Approximation Error and Bayes' Consistency</h4>
<ul>
<li>Nested Function Spaces</li>
<li>Regularization</li>
<li>Achieving Zero Approximation Error</li>
<li>Rate of Convergence</li>
</ul>
<h4>
<a id="no-free-lunch-theorem" class="anchor" href="#no-free-lunch-theorem" aria-hidden="true"><span aria-hidden="true" class="octicon octicon-link"></span></a>No Free Lunch Theorem</h4>
<ul>
<li>Algorithmic Consistency</li>
<li>NFT Formal Statements</li>
</ul>
<h3>
<a id="problem-space-and-algorthm-families" class="anchor" href="#problem-space-and-algorthm-families" aria-hidden="true"><span aria-hidden="true" class="octicon octicon-link"></span></a>Problem Space and Algorthm Families</h3>
<h4>
<a id="generative-and-discriminative-models" class="anchor" href="#generative-and-discriminative-models" aria-hidden="true"><span aria-hidden="true" class="octicon octicon-link"></span></a>Generative and Discriminative Models</h4>
<ul>
<li>Generative Models</li>
<li>Discriminant Models</li>
<li>Examples of Discriminant Approaches</li>
<li>Differences between Generative and Discriminant Models</li>
</ul>
<h4>
<a id="supervised-learning" class="anchor" href="#supervised-learning" aria-hidden="true"><span aria-hidden="true" class="octicon octicon-link"></span></a>Supervised Learning</h4>
<ul>
<li>Supervised Learning Practice Steps</li>
<li>Challenges with Supervised Learning Practice</li>
<li>Formulation</li>
</ul>
<h4>
<a id="unsupervised-learning" class="anchor" href="#unsupervised-learning" aria-hidden="true"><span aria-hidden="true" class="octicon octicon-link"></span></a>Unsupervised Learning</h4>
<h4>
<a id="machine-learning" class="anchor" href="#machine-learning" aria-hidden="true"><span aria-hidden="true" class="octicon octicon-link"></span></a>Machine Learning</h4>
<ul>
<li>Calibration vs. Learning</li>
</ul>
<h4>
<a id="pattern-recognition" class="anchor" href="#pattern-recognition" aria-hidden="true"><span aria-hidden="true" class="octicon octicon-link"></span></a>Pattern Recognition</h4>
<ul>
<li>Supervised vs. Unsupervised Pattern Recognition</li>
<li>Probabilistic Pattern Recognition</li>
<li>Formulation of Pattern Recognition</li>
<li>Pattern Recognition Practice SKU</li>
<li>Pattern Recognition Applications</li>
</ul>
<h3>
<a id="parametric-classification-algorithms" class="anchor" href="#parametric-classification-algorithms" aria-hidden="true"><span aria-hidden="true" class="octicon octicon-link"></span></a>Parametric Classification Algorithms</h3>
<h4>
<a id="statistical-classification" class="anchor" href="#statistical-classification" aria-hidden="true"><span aria-hidden="true" class="octicon octicon-link"></span></a>Statistical Classification</h4>
<h4>
<a id="linear-discriminant-analysis" class="anchor" href="#linear-discriminant-analysis" aria-hidden="true"><span aria-hidden="true" class="octicon octicon-link"></span></a>Linear Discriminant Analysis</h4>
<ul>
<li>Setup and Formulation</li>
<li>Fischer's Linear Discriminant</li>
<li>Quadratic Discriminant Analysis</li>
</ul>
<h4>
<a id="logistic-regression" class="anchor" href="#logistic-regression" aria-hidden="true"><span aria-hidden="true" class="octicon octicon-link"></span></a>Logistic Regression</h4>
<ul>
<li>Formulation</li>
<li>Goodness of Fit</li>
<li>Mathematical Setup</li>
<li>Bayesian Logistic Regression</li>
<li>Logistic Regression Extensions</li>
<li>Model Suitability Tests with Cross Validation</li>
</ul>
<h4>
<a id="multinomial-logistic-regression" class="anchor" href="#multinomial-logistic-regression" aria-hidden="true"><span aria-hidden="true" class="octicon octicon-link"></span></a>Multinomial Logistic Regression</h4>
<ul>
<li>Setup and Formulation</li>
</ul>
<h3>
<a id="non-parametric-classification-algorithms" class="anchor" href="#non-parametric-classification-algorithms" aria-hidden="true"><span aria-hidden="true" class="octicon octicon-link"></span></a>Non-Parametric Classification Algorithms</h3>
<h4>
<a id="decision-trees-and-decision-lists" class="anchor" href="#decision-trees-and-decision-lists" aria-hidden="true"><span aria-hidden="true" class="octicon octicon-link"></span></a>Decision Trees and Decision Lists</h4>
<h4>
<a id="variable-bandwidth-kernel-density-estimation" class="anchor" href="#variable-bandwidth-kernel-density-estimation" aria-hidden="true"><span aria-hidden="true" class="octicon octicon-link"></span></a>Variable Bandwidth Kernel Density Estimation</h4>
<h4>
<a id="k-nearest-neighbors-algorithm" class="anchor" href="#k-nearest-neighbors-algorithm" aria-hidden="true"><span aria-hidden="true" class="octicon octicon-link"></span></a>k Nearest Neighbors Algorithm</h4>
<h4>
<a id="perceptron" class="anchor" href="#perceptron" aria-hidden="true"><span aria-hidden="true" class="octicon octicon-link"></span></a>Perceptron</h4>
<h4>
<a id="support-vector-machines-svm" class="anchor" href="#support-vector-machines-svm" aria-hidden="true"><span aria-hidden="true" class="octicon octicon-link"></span></a>Support Vector Machines (SVM)</h4>
<h4>
<a id="gene-expression-programming-gep" class="anchor" href="#gene-expression-programming-gep" aria-hidden="true"><span aria-hidden="true" class="octicon octicon-link"></span></a>Gene Expression Programming (GEP)</h4>
<h3>
<a id="clustering-algorithms" class="anchor" href="#clustering-algorithms" aria-hidden="true"><span aria-hidden="true" class="octicon octicon-link"></span></a>Clustering Algorithms</h3>
<h4>
<a id="cluster-analysis" class="anchor" href="#cluster-analysis" aria-hidden="true"><span aria-hidden="true" class="octicon octicon-link"></span></a>Cluster Analysis</h4>
<ul>
<li>Cluster Models</li>
<li>Connectivity Based Clustering</li>
<li>Centroid Based Clustering</li>
<li>Distribution Based Clustering</li>
<li>Density Based Clustering</li>
<li>Clustering Enhancements</li>
<li>Internal Cluster Evaluation</li>
<li>External Cluster Evaluation</li>
<li>Clustering Axiom</li>
</ul>
<h4>
<a id="mixture-model" class="anchor" href="#mixture-model" aria-hidden="true"><span aria-hidden="true" class="octicon octicon-link"></span></a>Mixture Model</h4>
<ul>
<li>Generic Mixture Model Details</li>
<li>Specific Mixture Models</li>
<li>Mixture Model Samples</li>
<li>Identifiability</li>
<li>Expectation Maximization</li>
<li>Alternatives to Expectation Maximization</li>
<li>Mixture Model Extensions</li>
</ul>
<h4>
<a id="deep-learning" class="anchor" href="#deep-learning" aria-hidden="true"><span aria-hidden="true" class="octicon octicon-link"></span></a>Deep Learning</h4>
<ul>
<li>Unsupervised Representation Learner</li>
<li>Deep Learning using ANN</li>
<li>Deep Learning Architectures</li>
<li>Challenges with the DNN Approach</li>
<li>Deep Belief Networks (DBN)</li>
<li>Convolutional Neural Networks (CNN)</li>
<li>Deep Learning Evaluation Data Sets</li>
<li>Neurological Basis of Deep Learning</li>
</ul>
<h4>
<a id="hierarchical-clustering" class="anchor" href="#hierarchical-clustering" aria-hidden="true"><span aria-hidden="true" class="octicon octicon-link"></span></a>Hierarchical Clustering</h4>
<h4>
<a id="k-means-clustering" class="anchor" href="#k-means-clustering" aria-hidden="true"><span aria-hidden="true" class="octicon octicon-link"></span></a>k-Means Clustering</h4>
<ul>
<li>Mathematical Formulation</li>
<li>The Standard Algorithm</li>
<li>k-Means Initialization Schemes</li>
<li>k-Means Complexity</li>
<li>k-Means Variations</li>
<li>k-Means Applications</li>
<li>Alternate k-Means Formulations</li>
</ul>
<h4>
<a id="correlation-clustering" class="anchor" href="#correlation-clustering" aria-hidden="true"><span aria-hidden="true" class="octicon octicon-link"></span></a>Correlation Clustering</h4>
<h4>
<a id="kernel-principal-component-analysis-kernel-pca" class="anchor" href="#kernel-principal-component-analysis-kernel-pca" aria-hidden="true"><span aria-hidden="true" class="octicon octicon-link"></span></a>Kernel Principal Component Analysis (Kernel PCA)</h4>
<h3>
<a id="ensemble-learning-algorithms" class="anchor" href="#ensemble-learning-algorithms" aria-hidden="true"><span aria-hidden="true" class="octicon octicon-link"></span></a>Ensemble Learning Algorithms</h3>
<h4>
<a id="ensemble-learning" class="anchor" href="#ensemble-learning" aria-hidden="true"><span aria-hidden="true" class="octicon octicon-link"></span></a>Ensemble Learning</h4>
<ul>
<li>Overview</li>
<li>Theoretical Underpinnings</li>
<li>Ensemble Aggregator Types</li>
<li>Bayes' Optimal Classifier</li>
<li>Bagging and Boosting</li>
<li>Bayesian Model Averaging (BMA)</li>
<li>Baysian Model Combination (BMC)</li>
<li>Bucket of Models (BOM)</li>
<li>Stacking</li>
<li>Ensemble Averaging vs. Basis Spline Representation</li>
</ul>
<h4>
<a id="ann-ensemble-averaging" class="anchor" href="#ann-ensemble-averaging" aria-hidden="true"><span aria-hidden="true" class="octicon octicon-link"></span></a>ANN Ensemble Averaging</h4>
<ul>
<li>Techniques and Results</li>
</ul>
<h4>
<a id="boosting" class="anchor" href="#boosting" aria-hidden="true"><span aria-hidden="true" class="octicon octicon-link"></span></a>Boosting</h4>
<ul>
<li>Philosophy behind Boosting Algorithms</li>
<li>Popular Boosting Algorithms and Drawbacks</li>
</ul>
<h4>
<a id="bootstrap-averaging" class="anchor" href="#bootstrap-averaging" aria-hidden="true"><span aria-hidden="true" class="octicon octicon-link"></span></a>Bootstrap Averaging</h4>
<ul>
<li>Sample Generation</li>
<li>Bagging with 1NN - Theoretical Treatment</li>
</ul>
<h3>
<a id="multi-linear-sub-space-learning-algorithms" class="anchor" href="#multi-linear-sub-space-learning-algorithms" aria-hidden="true"><span aria-hidden="true" class="octicon octicon-link"></span></a>Multi-linear Sub-space Learning Algorithms</h3>
<h4>
<a id="tensors-and-multi-linear-sub-space-algorithms" class="anchor" href="#tensors-and-multi-linear-sub-space-algorithms" aria-hidden="true"><span aria-hidden="true" class="octicon octicon-link"></span></a>Tensors and Multi-linear Sub-space Algorithms</h4>
<ul>
<li>Tensors</li>
<li>Multi-linear Sub-space Learning</li>
<li>Multi-linear PCA</li>
</ul>
<h3>
<a id="real-valued-sequence-learning-algorithms" class="anchor" href="#real-valued-sequence-learning-algorithms" aria-hidden="true"><span aria-hidden="true" class="octicon octicon-link"></span></a>Real-Valued Sequence Learning Algorithms</h3>
<h4>
<a id="kalman-filtering" class="anchor" href="#kalman-filtering" aria-hidden="true"><span aria-hidden="true" class="octicon octicon-link"></span></a>Kalman Filtering</h4>
<ul>
<li>Continuous Time Kalman Filtering</li>
<li>Nonlinear Kalman Filtering</li>
<li>Kalman Smoothing</li>
</ul>
<h4>
<a id="particle-filtering" class="anchor" href="#particle-filtering" aria-hidden="true"><span aria-hidden="true" class="octicon octicon-link"></span></a>Particle Filtering</h4>
<h3>
<a id="real-valued-learning-algorithms" class="anchor" href="#real-valued-learning-algorithms" aria-hidden="true"><span aria-hidden="true" class="octicon octicon-link"></span></a>Real-Valued Learning Algorithms</h3>
<h4>
<a id="regression-analysis" class="anchor" href="#regression-analysis" aria-hidden="true"><span aria-hidden="true" class="octicon octicon-link"></span></a>Regression Analysis</h4>
<ul>
<li>Linear Regression</li>
<li>Assumptions Underlying Basis Linear Regression</li>
<li>Multi-variate Regression Analysis</li>
<li>Multi-variate Predictor/Response Regression</li>
<li>OLS on Basis Spline Representation</li>
<li>OLS on Basis Spline Representation with Roughness Penalty</li>
<li>Linear Regression Estimator Extensions</li>
<li>Bayesian Approach to Regression Analysis</li>
</ul>
<h4>
<a id="component-analysis" class="anchor" href="#component-analysis" aria-hidden="true"><span aria-hidden="true" class="octicon octicon-link"></span></a>Component Analysis</h4>
<ul>
<li>Independent Component Analysis (ICA) Specification</li>
<li>Independent Component Analysis (ICA) Formulation</li>
<li>Principal Component Analysis</li>
<li>Principal COmponent Analysis - Constrained Formulation</li>
<li>2D Principal Component Analysis - Constrained Formulation</li>
<li>2D Principal Component Analysis - Lagrange Multiplier Based Constrained Formulation</li>
<li>nD Principal Component Analysis - Lagrange Multiplier Based Constrained Formulation</li>
<li>Information Theoretic Analysis of PCA</li>
<li>Empirical PCA Estimation From Data Set</li>
</ul>
<h3>
<a id="sequence-label-learning-algorithms" class="anchor" href="#sequence-label-learning-algorithms" aria-hidden="true"><span aria-hidden="true" class="octicon octicon-link"></span></a>Sequence Label Learning Algorithms</h3>
<h4>
<a id="hidden-markov-models" class="anchor" href="#hidden-markov-models" aria-hidden="true"><span aria-hidden="true" class="octicon octicon-link"></span></a>Hidden Markov Models</h4>
<ul>
<li>HMM State Transition/Emission Parameter Estimation</li>
<li>HMM Based Inference</li>
<li>Non-Bayesian HMM Model Setup</li>
<li>Bayesian Extension to the HMM Model Setup</li>
<li>HMM in Practical World</li>
</ul>
<h4>
<a id="markov-chain-models" class="anchor" href="#markov-chain-models" aria-hidden="true"><span aria-hidden="true" class="octicon octicon-link"></span></a>Markov Chain Models</h4>
<ul>
<li>Markov Property</li>
<li>Markov Chains</li>
<li>Classification of the Markov Models</li>
<li>Monte Carlo Markov Chains (MCMC)</li>
<li>MCMC for Multi-dimensional Integrals</li>
</ul>
<h4>
<a id="markov-random-and-conditional-fields" class="anchor" href="#markov-random-and-conditional-fields" aria-hidden="true"><span aria-hidden="true" class="octicon octicon-link"></span></a>Markov Random and Conditional Fields</h4>
<ul>
<li>MRF/CRF Axiomatic Properties/Definitions</li>
<li>Clique Factorization</li>
<li>Inference in MRF/CRF</li>
</ul>
<h4>
<a id="maximum-entropy-markov-models" class="anchor" href="#maximum-entropy-markov-models" aria-hidden="true"><span aria-hidden="true" class="octicon octicon-link"></span></a>Maximum Entropy Markov Models</h4>
<h4>
<a id="probabilistic-grammar-and-parsing" class="anchor" href="#probabilistic-grammar-and-parsing" aria-hidden="true"><span aria-hidden="true" class="octicon octicon-link"></span></a>Probabilistic Grammar and Parsing</h4>
<ul>
<li>Parsing</li>
<li>Parser</li>
<li>Context-Free Grammar (CFG)</li>
</ul>
<h3>
<a id="bayesian-analysis" class="anchor" href="#bayesian-analysis" aria-hidden="true"><span aria-hidden="true" class="octicon octicon-link"></span></a>Bayesian Analysis</h3>
<h4>
<a id="concepts-formulation-usage-and-application" class="anchor" href="#concepts-formulation-usage-and-application" aria-hidden="true"><span aria-hidden="true" class="octicon octicon-link"></span></a>Concepts, Formulation, Usage, and Application</h4>
<ul>
<li>Applicability</li>
<li>Analysis of Bayesian Systems</li>
<li>Bayesian Networks</li>
<li>Hypothesis Testing</li>
<li>Bayesian Updating</li>
<li>Maximum Entropy Techniques</li>
<li>Priors</li>
<li>Predictive Posteriors and Priors</li>
<li>Approximate Bayesian Computation</li>
<li>Measurement and Parametric Calibration</li>
<li>Bayesian Regression Analysis</li>
<li>Extensions to Bayesian Regression Analysis</li>
<li>Spline Proxying of Bayesian Systems</li>
</ul>
<h3>
<a id="linear-algebra-support" class="anchor" href="#linear-algebra-support" aria-hidden="true"><span aria-hidden="true" class="octicon octicon-link"></span></a>Linear Algebra Support</h3>
<h4>
<a id="optimizer" class="anchor" href="#optimizer" aria-hidden="true"><span aria-hidden="true" class="octicon octicon-link"></span></a>Optimizer</h4>
<ul>
<li>Constrained Optimization using Lagrangian</li>
<li>Least Squares Optimizer</li>
<li>Multi-variate Distribution</li>
</ul>
<h4>
<a id="linear-systems-analysis-and-transformation" class="anchor" href="#linear-systems-analysis-and-transformation" aria-hidden="true"><span aria-hidden="true" class="octicon octicon-link"></span></a>Linear Systems Analysis and Transformation</h4>
<ul>
<li>Matrix Transforms</li>
<li>System of Linear Equations</li>
<li>Orthogonalization</li>
<li>Gaussian Elimination</li>
</ul>
<h4>
<a id="rayleigh-quotient-iteration" class="anchor" href="#rayleigh-quotient-iteration" aria-hidden="true"><span aria-hidden="true" class="octicon octicon-link"></span></a>Rayleigh Quotient Iteration</h4>
<ul>
<li>Introduction</li>
<li>The Algorithm</li>
<li>References</li>
</ul>
<h4>
<a id="power-iteration" class="anchor" href="#power-iteration" aria-hidden="true"><span aria-hidden="true" class="octicon octicon-link"></span></a>Power Iteration</h4>
<ul>
<li>Introduction</li>
<li>The Method</li>
<li>Analysis</li>
<li>Applications</li>
<li>References</li>
</ul>
<h2>
<a id="contact" class="anchor" href="#contact" aria-hidden="true"><span aria-hidden="true" class="octicon octicon-link"></span></a>Contact</h2>
<p><a href="mailto:[email protected]">[email protected]</a></p>
</section>
</div>
<footer>
<p>Project maintained by <a href="https://github.com/lakshmiDRIP">lakshmiDRIP</a></p>
<p>Hosted on GitHub Pages — Theme by <a href="https://github.com/orderedlist">orderedlist</a></p>
</footer>
<!--[if !IE]><script>fixScale(document);</script><![endif]-->
</body>
</html>