From 7d7b1047a853f8708aa3b0f336db5d98acae327e Mon Sep 17 00:00:00 2001 From: Juliette Moreau Date: Fri, 27 Oct 2023 17:02:33 +0200 Subject: [PATCH 01/12] creation des fichiers --- collections/_posts/2023-10-27-LUPI.md | 46 ++++++++++++++++++ .../images/LUPI/comparaison_SVM_Bayes.png | Bin 0 -> 66576 bytes .../LUPI/error_rate_Mackey-Glass_series.png | Bin 0 -> 47403 bytes .../LUPI/error_rates_digit_recognition.png | Bin 0 -> 66360 bytes .../LUPI/error_rates_protein_folding.png | Bin 0 -> 58916 bytes 5 files changed, 46 insertions(+) create mode 100644 collections/_posts/2023-10-27-LUPI.md create mode 100644 collections/images/LUPI/comparaison_SVM_Bayes.png create mode 100644 collections/images/LUPI/error_rate_Mackey-Glass_series.png create mode 100644 collections/images/LUPI/error_rates_digit_recognition.png create mode 100644 collections/images/LUPI/error_rates_protein_folding.png diff --git a/collections/_posts/2023-10-27-LUPI.md b/collections/_posts/2023-10-27-LUPI.md new file mode 100644 index 00000000..4f3ef4e7 --- /dev/null +++ b/collections/_posts/2023-10-27-LUPI.md @@ -0,0 +1,46 @@ +--- +layout: review +title: "LUPI" +tags: machine learning, SVM, privileged information +author: "Juliette Moreau" +cite: + authors: "Vladimir Vapnik and Akshay Vashist" + title: "A new learning paradigm: learning using privileged information" + venue: "Neural Networks, 2009" +pdf: "https://arxiv.org/pdf/1905.03677.pdf" +--- + + +# Highlights + +* This paper introduces to a new leanrning paradigm called LUPI. +* The addition of privileged information during the training even if it is not during the inference improves the performances. +* The paradigm is tested on three different applications. + +# Introduction + + + +# Problematization + + + +# Privileged information in SVM type algorithms: the SVM+ method + + + +# Three examples of privileged information + +## Advanced technival model as privileged information: protein folding + + +## Future events as privileged information: Mackey-Glass series + + +## Hollistic description as privileged information: digits classification + + + +# Discussion + + diff --git a/collections/images/LUPI/comparaison_SVM_Bayes.png b/collections/images/LUPI/comparaison_SVM_Bayes.png new file mode 100644 index 0000000000000000000000000000000000000000..1cbd4cbb49af17ec24d4f273e9447dfde95d739d GIT binary patch literal 66576 zcmafbcU+He`+jypMo}mYl_a!DDQy+nJEFZ5ZCa!urCo}Yrcy~mLwnE=B}z+!5}GRQ z@jI^1^ZNe%`~2~EyuHVLU)On^=W!h8ao+yQm*jR(9iXD1px7aQURsrcVqHE3#fBY} zoAJt|&D~b~vC&>qUY!#ExltP5$KUrm$Y?vL*_t{yU$ZlzFtf3}ZgSk-$j-#X#@^i4 zVPZpt7~aH5-Xvvba?QcQ)`nT#;<^cilZnjl-~;>-*+?O4CSeY|Ut|>u{&{XibmE zbf9Dy+g6EP8zt(02n%S6B>oZM{hTflgYI1}{%-3T?q3-fsW@e`e9@ z$TQpd+OkbjNhw(OwI#3FM+QEA{az9Rt%2%I z2ZXHCT06vshTcEoq@~Kw&u?jOPj&hGy`!`9=8ty|o8w%^x&%2c#m;}=rIfscYYVw7 zExmEW7HZ=vuT8bJwXY1{QWaQtR_B_$AEZ7W&|Bs@Jv*k?TkifoSw8&N*X;P*v)8J< ztqq=QX6aggd-E@^?(!Rlt&EIMZMO;s%+I^De*WzL;6b9=^GgD@y|G7>;@qZU&6?wG 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He/she provides additional information hidden in explanations, comments or comparisons. The paper explores a new paradigm in which specific forms of privileged information are considered during the training but not during the test phase. The superiority of this paradigm over the classical one is demonstrated. it is called the LUPI paradigm for learning using priviledge information. # Problematization +Usually the training is based on a set of pairs: + +$$(x_1,y_1),(x_2, y_2),...,(x_n,y_n) \quad where \quad x_i \in X, \quad y_i \in \{-1,1\} $$ + +following an unknown probability measure $$P(x,y)$$ and the goal is to find among a collection of functions $$f(x, \alpha), \alpha \in \Lambda$$ the function $$y=f(x, \alpha*)$$ that minimizes the number of incorrect classifications. + +In the LUPI paradigm we have triplets: + +$$(x_1, x^*_1, y_1),(x_2, x^*_2, y_2),...,(x_n, x^*_n, y_n) \quad where \quad x_i \in X, \quad x^*_i \in X^*, \quad y_i \in \{-1,1\} $$ +following the unknow probabilty measure $$P(x, x^*, y)$$ and the goal is to find among a collection of functions $$f(x, \alpha), \alpha \in \Lambda$$ the function $$y=f(x, \alpha*)$$ that minimizes the number of incorrect classifications. +The objective is exactly the same, but during training there is additional information that is not available during the test, that's why it's privileged information. + +They imagine several types of privileged information and some examples in the medical domain. For temporal prediction, some information from intermediate steps can be added (knowing the patient's symptoms at 3, 6 and 9 months to predict the evolution at 12 months). For images, holistic descriptions can be used (doctor's description of biopsy to classify between healthy and cancerous). # Privileged information in SVM type algorithms: the SVM+ method +They apply this paradigm to SVM, the goal is to show that the convergence rate is significantly improved with this method. + +Some propositions are demontrated introducing oracle SVM. + +> Proposition 1: If any vector $$x \in X$$ belongs to one and only one of the classes and there exists an Oracle function with respect to the best decision rule in the asmissible set of hyperparameters, then with the probablity $$-\eta$$ the following bound holds true +> +> $$ P(y[(w_l,x)+b_l]<0) \leq P(1-\xi^0 <0) + A \frac{h ln\frac{l}{h}-ln (\eta)}{l}$$ +> +> where $$P(y[(w_l,x)+b_l]<0)$$ is the probability of error for the Oracle SVM solution on the training set of size $$l$$, $$P(1-\xi^0 <0)$$ is the probability of error for the best solution in the asmissible set of functions, $$h$$ is the VC dimension of the admissible set of hyperplanes, ans $$A$$ is a constant. +> +> That is the Oracle solution converges to the best possible solution in the admissible set of solutions with the rate $$O(h/l)$$. + +But in reality a teacher does not know the oracle function, but can supply privileged information instead with the admissible set of the correcting functions $$\phi(x^*, \delta), \delta \in \Delta$$ whiche defines that values of the oracle function $$ \xi^0(x_i) = \phi(x^*_i, \delta_0), \forall (x_i, x^*_i, y_i)$$ + +![](/collections/images/LUPI/comparaison_SVM_Bayes.jpg) +> Proposition 2: Under the conditions of proposition 1 with probablity $$1-\eta$$ the following bound holds true +> +> $$ P(y[(w_l,x)+b_l]<0) \leq P(1-\phi(x^*, \delta_l<0)+A \frac{(h+h^*) ln\frac{2l}{h+h^*}-ln (\eta)}{l}$$ +> +> where $$P(y[(w_l,x)+b_l]<0)$$ is the probability of error for solution of the problem $$R(w,b,\delta) = \sum_{i=1}^{l} \theta [\phi(x^*_i, \delta)-1]$$ (where $$\theta(u)=1$$ if $$u>0$$) subjects to constraints $$y_i[(w,x_i)+b] \geq -\phi(x^*_i, \delta), i=1,\ldots,l$$ on the training set of size $$l$$, $$P(1-\phi(x^*, \delta_l<0)$$ is the probability of event $$\{\phi(x^*, \delta_l)>1\}$$, $$h$$ is the VC dimension of the admissible set of hyperplanes and $$h^*$$ is the VC dimension of the admissible set of correcting functions. + +For the rest, two models of the set of correcting functions are considered: the general $$X^*$$SVM+ model (an admissible set of non-negative correcting functions is defined in the multi-dimensional $$X^*$$-space) and the particular dSVM+ model (a set of admissible non-negative correcting functions in defined in a special one-dimensional d-space). The "plus" designs the use of privilged information. # Three examples of privileged information -## Advanced technival model as privileged information: protein folding +## Advanced technical model as privileged information: protein folding + +Proteins are classified in a human-made hierarchical classification according to their 3D structure in order to know how they are connected through evolution. But this 3D structure is complicated and time-consuming to obtain, whereas the amino acid sequence is easy to obtain, but getting the first from the second is not straightforward. + +The aim is to classify the proteins in this classification, but only with the amino acid sequence thanks to a pattern recognition technique. SVMs are considered to be one of the best techniques for constructing this type of decision rule. + +They focus on finding homologies between amino acid sequences between superfamilies in 80 cases of binary classification with 80 superfamilies that are representative of protein diversity and have enough sequences to be significant. Similarity between sequences is computed with profile-kernel, similarity between 3D structures with MAMMOTH. The set of vectors $$x \in X$$ is the similarity between sequences used for training, while $$x^* \in X^*$$ is the similarity between 3D structures used as privileged information. + +![](/collections/images/LUPI/error_rates_protein_folding.jpg) +The classical paradigm is in no way superior to the LUPI paradigm. In 3 cases there is no error for either paradigm. In 11 cases there is no difference between them. In all other cases LUPI improves the results. dSVM+ is better than $$X^*$$SVM+, but the classification with the 3D structure remains better. ## Future events as privileged information: Mackey-Glass series +There are two parametrisations of such problem: quantitatively when with the value at time $$t$$ we predict the value at $$t+Dt$$ or qualitatively if with the value at time $$t$$ we estimate if it is greater of lower at time $$t+Dt$$. + +In the first case LUPI can be used with a regression model and in the second case a pattern recognition model can be used. The latter case is studied in the article using chaotic series and the Mackey-Glass equation. These equations are used in several models and one in particular, which shows that the SVM is better than many other techniques for predicting time series, is used as a comparison for the LUPI paradigm. + +The goal is to predict whether $$x(t+T)>x(t)$$ with a 4-dimensional vector of time series observations: only one step ahead, or also 5 and 8 steps ahead of the time of interest. + +![](/collections/images/LUPI/error_rate_Mackey-Glass_series.jpg) + +The bayesian law is approximated to evaluate the poximity between the error of the Oracle SVM and the bayesian error considering that the SVM error converge to bayesian error. + ## Hollistic description as privileged information: digits classification +As an example of holistic description as privileged information, they use the MNIST classification between 5 and 8 with resized to 10x10 images. A poetic description of the numbers is produced and translated into a 21-dimensional feature vector that gives characteristics of the number. +![](/collections/images/LUPI/error_rates_digit_recognition.jpg) # Discussion +3D information is very strong, so classification based on it remains better, but there are numerous cases where the LUPI paradigm gives remarkable results, such as with time series. 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zXRD^B*56c}@ZUap_+0NkzKdov(iH%QvRI5o+t@fKE?TqQe_MU)7smsb%P()Le4?Ce8;tD{e_8cWqon3dn>SAq z;UXNm9q#%3^GpBXwAvXla|GP%mR`7c64Qft%2&sEpv6f}PX4suu+;d4w(7m{wzJtT z?oTRsmYR?t&(YB9;yecjE&anGze|nhW0X~|{a1A;Z~VD-Li>IH{pn@x*7n}NzvbxX zQ$9)l|Mwd@^#A`4% Date: Tue, 31 Oct 2023 08:25:28 +0100 Subject: [PATCH 04/12] corrections proposed by Sophie --- collections/_posts/2023-10-27-LUPI.md | 14 +++++++------- 1 file changed, 7 insertions(+), 7 deletions(-) diff --git a/collections/_posts/2023-10-27-LUPI.md b/collections/_posts/2023-10-27-LUPI.md index 188c11be..907ef1f0 100644 --- a/collections/_posts/2023-10-27-LUPI.md +++ b/collections/_posts/2023-10-27-LUPI.md @@ -12,8 +12,8 @@ cite: # Highlights -* This paper introduces to a new leanrning paradigm called LUPI. -* The addition of privileged information during the training even if it is not during the inference improves the performances. +* This paper introduces a new learning paradigm called LUPI. +* The addition of privileged information during the training even if it is not present during the inference improves the performances. * The paradigm is tested on three different applications. # Introduction @@ -31,7 +31,7 @@ In the LUPI paradigm we have triplets: $$(x_1, x^*_1, y_1),(x_2, x^*_2, y_2),...,(x_n, x^*_n, y_n) \quad where \quad x_i \in X, \quad x^*_i \in X^*, \quad y_i \in \{-1,1\} $$ -following the unknow probabilty measure $$P(x, x^*, y)$$ and the goal is to find among a collection of functions $$f(x, \alpha), \alpha \in \Lambda$$ the function $$y=f(x, \alpha*)$$ that minimizes the number of incorrect classifications. +following the unknown probability measure $$P(x, x^*, y)$$ and the goal is to find among a collection of functions $$f(x, \alpha), \alpha \in \Lambda$$ the function $$y=f(x, \alpha*)$$ that minimizes the number of incorrect classifications. The objective is exactly the same, but during training there is additional information that is not available during the test, that's why it's privileged information. @@ -43,15 +43,15 @@ They apply this paradigm to SVM, the goal is to show that the convergence rate i Some propositions are demontrated introducing oracle SVM. -> Proposition 1: If any vector $$x \in X$$ belongs to one and only one of the classes and there exists an Oracle function with respect to the best decision rule in the asmissible set of hyperparameters, then with the probablity $$-\eta$$ the following bound holds true +> Proposition 1: If any vector $$x \in X$$ belongs to one and only one of the classes and there exists an Oracle function with respect to the best decision rule in the admissible set of hyperparameters, then with the probablity $$-\eta$$ the following bound holds true > > $$ P(y[(w_l,x)+b_l]<0) \leq P(1-\xi^0 <0) + A \frac{h ln\frac{l}{h}-ln (\eta)}{l}$$ > -> where $$P(y[(w_l,x)+b_l]<0)$$ is the probability of error for the Oracle SVM solution on the training set of size $$l$$, $$P(1-\xi^0 <0)$$ is the probability of error for the best solution in the asmissible set of functions, $$h$$ is the VC dimension of the admissible set of hyperplanes, ans $$A$$ is a constant. +> where $$P(y[(w_l,x)+b_l]<0)$$ is the probability of error for the Oracle SVM solution on the training set of size $$l$$, $$P(1-\xi^0 <0)$$ is the probability of error for the best solution in the admissible set of functions, $$h$$ is the VC dimension of the admissible set of hyperplanes, ans $$A$$ is a constant. > > That is the Oracle solution converges to the best possible solution in the admissible set of solutions with the rate $$O(h/l)$$. -But in reality a teacher does not know the oracle function, but can supply privileged information instead with the admissible set of the correcting functions $$\phi(x^*, \delta), \delta \in \Delta$$ whiche defines that values of the oracle function $$ \xi^0(x_i) = \phi(x^*_i, \delta_0), \forall (x_i, x^*_i, y_i)$$ +But in reality a teacher does not know the oracle function, but can supply privileged information instead with the admissible set of the correcting functions $$\phi(x^*, \delta), \delta \in \Delta$$ which defines that values of the oracle function $$ \xi^0(x_i) = \phi(x^*_i, \delta_0), \forall (x_i, x^*_i, y_i)$$ ![](/collections/images/LUPI/comparaison_SVM_Bayes.jpg) @@ -79,7 +79,7 @@ The classical paradigm is in no way superior to the LUPI paradigm. In 3 cases th ## Future events as privileged information: Mackey-Glass series -There are two parametrisations of such problem: quantitatively when with the value at time $$t$$ we predict the value at $$t+Dt$$ or qualitatively if with the value at time $$t$$ we estimate if it is greater of lower at time $$t+Dt$$. +There are two parametrisations of such problem: quantitatively when with the value at time $$t$$ we predict the value at $$t+Dt$$ or qualitatively , when with the value at time $$t$$ we estimate if it is greater of lower at time $$t+Dt$$. In the first case LUPI can be used with a regression model and in the second case a pattern recognition model can be used. The latter case is studied in the article using chaotic series and the Mackey-Glass equation. These equations are used in several models and one in particular, which shows that the SVM is better than many other techniques for predicting time series, is used as a comparison for the LUPI paradigm. From 385b90194498fbdabf58cea4e9fd918ddd0b115f Mon Sep 17 00:00:00 2001 From: JulietteMoreau <71698524+JulietteMoreau@users.noreply.github.com> Date: Thu, 2 Nov 2023 10:30:36 +0100 Subject: [PATCH 05/12] Update collections/_posts/2023-10-27-LUPI.md Co-authored-by: ThierryJudge --- collections/_posts/2023-10-27-LUPI.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/collections/_posts/2023-10-27-LUPI.md b/collections/_posts/2023-10-27-LUPI.md index 907ef1f0..eae16bf7 100644 --- a/collections/_posts/2023-10-27-LUPI.md +++ b/collections/_posts/2023-10-27-LUPI.md @@ -17,7 +17,7 @@ cite: * The paradigm is tested on three different applications. # Introduction -Unlike machine learning, where the role of the teacher is not important, in human learning the teacher is very important. He/she provides additional information hidden in explanations, comments or comparisons. The paper explores a new paradigm in which specific forms of privileged information are considered during the training but not during the test phase. The superiority of this paradigm over the classical one is demonstrated. it is called the LUPI paradigm for learning using priviledge information. +Unlike machine learning, where the role of the teacher is not important, in human learning the teacher is very important. He/she provides additional information hidden in explanations, comments or comparisons. The paper explores a new paradigm in which specific forms of privileged information are considered during the training but not during the test phase. The superiority of this paradigm over the classical one is demonstrated. it is called the LUPI paradigm for learning using privileged information. # Problematization From 912a925a410325c1388d3f490592ed4c2048bfc7 Mon Sep 17 00:00:00 2001 From: JulietteMoreau <71698524+JulietteMoreau@users.noreply.github.com> Date: Thu, 2 Nov 2023 10:30:52 +0100 Subject: [PATCH 06/12] Update collections/_posts/2023-10-27-LUPI.md Co-authored-by: ThierryJudge --- collections/_posts/2023-10-27-LUPI.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/collections/_posts/2023-10-27-LUPI.md b/collections/_posts/2023-10-27-LUPI.md index eae16bf7..a67a5df8 100644 --- a/collections/_posts/2023-10-27-LUPI.md +++ b/collections/_posts/2023-10-27-LUPI.md @@ -41,7 +41,7 @@ They imagine several types of privileged information and some examples in the me They apply this paradigm to SVM, the goal is to show that the convergence rate is significantly improved with this method. -Some propositions are demontrated introducing oracle SVM. +Some propositions are demonstrated introducing oracle SVM. > Proposition 1: If any vector $$x \in X$$ belongs to one and only one of the classes and there exists an Oracle function with respect to the best decision rule in the admissible set of hyperparameters, then with the probablity $$-\eta$$ the following bound holds true > From 095b90184e6d067b5f4dfe9f217971ae912d71af Mon Sep 17 00:00:00 2001 From: JulietteMoreau <71698524+JulietteMoreau@users.noreply.github.com> Date: Thu, 2 Nov 2023 10:31:12 +0100 Subject: [PATCH 07/12] Update collections/_posts/2023-10-27-LUPI.md Co-authored-by: ThierryJudge --- collections/_posts/2023-10-27-LUPI.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/collections/_posts/2023-10-27-LUPI.md b/collections/_posts/2023-10-27-LUPI.md index a67a5df8..da677880 100644 --- a/collections/_posts/2023-10-27-LUPI.md +++ b/collections/_posts/2023-10-27-LUPI.md @@ -43,7 +43,7 @@ They apply this paradigm to SVM, the goal is to show that the convergence rate i Some propositions are demonstrated introducing oracle SVM. -> Proposition 1: If any vector $$x \in X$$ belongs to one and only one of the classes and there exists an Oracle function with respect to the best decision rule in the admissible set of hyperparameters, then with the probablity $$-\eta$$ the following bound holds true +> Proposition 1: If any vector $$x \in X$$ belongs to one and only one of the classes and there exists an Oracle function with respect to the best decision rule in the admissible set of hyperparameters, then with the probability $$-\eta$$ the following bound holds true > > $$ P(y[(w_l,x)+b_l]<0) \leq P(1-\xi^0 <0) + A \frac{h ln\frac{l}{h}-ln (\eta)}{l}$$ > From ccbb87176b799c764b2780415de905f390d0b1f2 Mon Sep 17 00:00:00 2001 From: JulietteMoreau <71698524+JulietteMoreau@users.noreply.github.com> Date: Thu, 2 Nov 2023 10:31:24 +0100 Subject: [PATCH 08/12] Update collections/_posts/2023-10-27-LUPI.md Co-authored-by: ThierryJudge --- collections/_posts/2023-10-27-LUPI.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/collections/_posts/2023-10-27-LUPI.md b/collections/_posts/2023-10-27-LUPI.md index da677880..3149f53e 100644 --- a/collections/_posts/2023-10-27-LUPI.md +++ b/collections/_posts/2023-10-27-LUPI.md @@ -47,7 +47,7 @@ Some propositions are demonstrated introducing oracle SVM. > > $$ P(y[(w_l,x)+b_l]<0) \leq P(1-\xi^0 <0) + A \frac{h ln\frac{l}{h}-ln (\eta)}{l}$$ > -> where $$P(y[(w_l,x)+b_l]<0)$$ is the probability of error for the Oracle SVM solution on the training set of size $$l$$, $$P(1-\xi^0 <0)$$ is the probability of error for the best solution in the admissible set of functions, $$h$$ is the VC dimension of the admissible set of hyperplanes, ans $$A$$ is a constant. +> where $$P(y[(w_l,x)+b_l]<0)$$ is the probability of error for the Oracle SVM solution on the training set of size $$l$$, $$P(1-\xi^0 <0)$$ is the probability of error for the best solution in the admissible set of functions, $$h$$ is the VC dimension of the admissible set of hyperplanes, and $$A$$ is a constant. > > That is the Oracle solution converges to the best possible solution in the admissible set of solutions with the rate $$O(h/l)$$. From 8daca5a6812e7a611a5ed86aa8e535865fc6ab47 Mon Sep 17 00:00:00 2001 From: JulietteMoreau <71698524+JulietteMoreau@users.noreply.github.com> Date: Thu, 2 Nov 2023 10:31:37 +0100 Subject: [PATCH 09/12] Update collections/_posts/2023-10-27-LUPI.md Co-authored-by: ThierryJudge --- collections/_posts/2023-10-27-LUPI.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/collections/_posts/2023-10-27-LUPI.md b/collections/_posts/2023-10-27-LUPI.md index 3149f53e..ef64dbef 100644 --- a/collections/_posts/2023-10-27-LUPI.md +++ b/collections/_posts/2023-10-27-LUPI.md @@ -55,7 +55,7 @@ But in reality a teacher does not know the oracle function, but can supply privi ![](/collections/images/LUPI/comparaison_SVM_Bayes.jpg) -> Proposition 2: Under the conditions of proposition 1 with probablity $$1-\eta$$ the following bound holds true +> Proposition 2: Under the conditions of proposition 1 with probability $$1-\eta$$ the following bound holds true > > $$ P(y[(w_l,x)+b_l]<0) \leq P(1-\phi(x^*, \delta_l<0)+A \frac{(h+h^*) ln\frac{2l}{h+h^*}-ln (\eta)}{l}$$ > From ad9f2ca0bc7fe8a1ad2d748b18a79d24c14b8778 Mon Sep 17 00:00:00 2001 From: JulietteMoreau <71698524+JulietteMoreau@users.noreply.github.com> Date: Thu, 2 Nov 2023 10:31:55 +0100 Subject: [PATCH 10/12] Update collections/_posts/2023-10-27-LUPI.md Co-authored-by: ThierryJudge --- collections/_posts/2023-10-27-LUPI.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/collections/_posts/2023-10-27-LUPI.md b/collections/_posts/2023-10-27-LUPI.md index ef64dbef..2979461f 100644 --- a/collections/_posts/2023-10-27-LUPI.md +++ b/collections/_posts/2023-10-27-LUPI.md @@ -61,7 +61,7 @@ But in reality a teacher does not know the oracle function, but can supply privi > > where $$P(y[(w_l,x)+b_l]<0)$$ is the probability of error for solution of the problem $$R(w,b,\delta) = \sum_{i=1}^{l} \theta [\phi(x^*_i, \delta)-1]$$ (where $$\theta(u)=1$$ if $$u>0$$) subjects to constraints $$y_i[(w,x_i)+b] \geq -\phi(x^*_i, \delta), i=1,\ldots,l$$ on the training set of size $$l$$, $$P(1-\phi(x^*, \delta_l<0)$$ is the probability of event $$\{\phi(x^*, \delta_l)>1\}$$, $$h$$ is the VC dimension of the admissible set of hyperplanes and $$h^*$$ is the VC dimension of the admissible set of correcting functions. -For the rest, two models of the set of correcting functions are considered: the general $$X^*$$SVM+ model (an admissible set of non-negative correcting functions is defined in the multi-dimensional $$X^*$$-space) and the particular dSVM+ model (a set of admissible non-negative correcting functions in defined in a special one-dimensional d-space). The "plus" designs the use of privilged information. +For the rest, two models of the set of correcting functions are considered: the general $$X^*$$SVM+ model (an admissible set of non-negative correcting functions is defined in the multi-dimensional $$X^*$$-space) and the particular dSVM+ model (a set of admissible non-negative correcting functions is defined in a special one-dimensional d-space). The "plus" designs the use of privileged information. # Three examples of privileged information From 1a6b180b0a6a368c1ec8f57e6102837262896c64 Mon Sep 17 00:00:00 2001 From: JulietteMoreau <71698524+JulietteMoreau@users.noreply.github.com> Date: Thu, 2 Nov 2023 10:32:14 +0100 Subject: [PATCH 11/12] Update collections/_posts/2023-10-27-LUPI.md Co-authored-by: ThierryJudge --- collections/_posts/2023-10-27-LUPI.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/collections/_posts/2023-10-27-LUPI.md b/collections/_posts/2023-10-27-LUPI.md index 2979461f..f6baea2f 100644 --- a/collections/_posts/2023-10-27-LUPI.md +++ b/collections/_posts/2023-10-27-LUPI.md @@ -79,7 +79,7 @@ The classical paradigm is in no way superior to the LUPI paradigm. In 3 cases th ## Future events as privileged information: Mackey-Glass series -There are two parametrisations of such problem: quantitatively when with the value at time $$t$$ we predict the value at $$t+Dt$$ or qualitatively , when with the value at time $$t$$ we estimate if it is greater of lower at time $$t+Dt$$. +There are two parametrizations of such problem: quantitatively when with the value at time $$t$$ we predict the value at $$t+Dt$$ or qualitatively , when with the value at time $$t$$ we estimate if it is greater or lower at time $$t+Dt$$. In the first case LUPI can be used with a regression model and in the second case a pattern recognition model can be used. The latter case is studied in the article using chaotic series and the Mackey-Glass equation. These equations are used in several models and one in particular, which shows that the SVM is better than many other techniques for predicting time series, is used as a comparison for the LUPI paradigm. From 2d1f2fd9c03725524596a24c36c4961d68e88170 Mon Sep 17 00:00:00 2001 From: JulietteMoreau <71698524+JulietteMoreau@users.noreply.github.com> Date: Thu, 2 Nov 2023 10:32:29 +0100 Subject: [PATCH 12/12] Update collections/_posts/2023-10-27-LUPI.md Co-authored-by: ThierryJudge --- collections/_posts/2023-10-27-LUPI.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/collections/_posts/2023-10-27-LUPI.md b/collections/_posts/2023-10-27-LUPI.md index f6baea2f..51a5389f 100644 --- a/collections/_posts/2023-10-27-LUPI.md +++ b/collections/_posts/2023-10-27-LUPI.md @@ -87,7 +87,7 @@ The goal is to predict whether $$x(t+T)>x(t)$$ with a 4-dimensional vector of ti ![](/collections/images/LUPI/error_rate_Mackey-Glass_series.jpg) -The bayesian law is approximated to evaluate the poximity between the error of the Oracle SVM and the bayesian error considering that the SVM error converge to bayesian error. +The Bayesian law is approximated to evaluate the proximity between the error of the Oracle SVM and the Bayesian error considering that the SVM error converges to Bayesian error. ## Hollistic description as privileged information: digits classification