A technique solves an SVM problem on table J, defined as the join of two tables T1 and T2, without explicitly joining the tables T1 and T2, in which the table T1 has m rows (piT, uiT), i=1, . . . , m, and the table T2 has n rows (qjT, vjT), j=1, . . . , n. A computer obtains a modified optimization problem from a primal optimization problem in which the modified optimization problem includes minimizew,b,η,ζ½∥w∥2+C·Σi=1mJ(i)·ηi+C·Σj=1nI(j)·ζj, subject to yixijTw−yib+ηi+ζj≧1 ((i,j)∈IJ) and ηi, ζj≧0. The penalty variables are reduced in the modified optimization problem by replacing the penalty variables in a form of ξij for each (i,j)∈IJ with the penalty variables in a form of ζij=ηi+ζj. A compact form of the modified optimization problem is obtained which includes minimizew,b,η,ζ,σ,τ½∥wP∥2+½∥wU∥2+½∥wQ∥2+C·Σi=1mJ(i)·ηi+C·Σj=1nI(j) ·ζj which is subject to yipiTwP−yib+ξi−σk≧0 (i∈Ik, k=1, . . . l), qjTwQ−τk≧0 (j∈Jk, k=1, . . . l), σk+zkTwU+τk≧1 (for k=1, . . . l such that Jk≠), σkzkTwU≧1 (for k=1, . . . l such that Jk=), and ξi≧0 (i=1, . . . , m). The compact form of the modified optimization problem is solved.
Please help me write a proper abstract based on the patent claims. CLAIM: 1. A method, by a computer, of solving a support vector machine problem on table J, defined as the join of two tables T1 and T2, without explicitly joining the tables T1 and T2, wherein the table T1 has m rows (piT, uiT), i=1, . . . , m, and the table T2 has n rows (qjT, vjT), j=1, . . . , n, the method comprising: providing a primal optimization problem over a join of the tables T1 and T2; obtaining, by the computer, a modified optimization problem from the primal optimization problem; reducing penalty variables in the modified optimization problem by replacing the penalty variables in a form of ξij for each (i,j)∈IJ with the penalty variables in a form of ξij=ηi+ζj; obtaining a compact form of the modified optimization problem in which the compact form comprises the penalty variables in the form of ξijηiζj; and solving the compact form of the modified optimization problem. 2. The method of claim 1, wherein the compact form comprises: Minimizew,b,η,ζ,σ,τ½∥wP∥2+½∥wU∥2½∥wQ∥2+C·Σi=1mJ(i)·ηi+C·Σj=1nI(j)·ζj, subject to yipiTwP−yib+ξi−σk≧0 (i∈Ik, k=1, . . . l) qjTwQ−τk≧0 (j∈Jk, k=1, . . . l) σk+zkTwU+τk≧1 (for k=1, . . . l such that Jk≠) σk+zkTwU≧1 (for k=1, . . . l such that Jk=) ξi≧0 (i=1, . . . , m); wherein the compact form includes auxiliary variables σ1, . . . , σl and τk for k=1, . . . l such that Jk≠. 3. The method of claim 2, wherein the primal optimization problem comprises: minimizew,b,ξ½∥w∥2+C·Σ(i,j)∈IJξij subject to yixijTw−yib+ξij≧1 ((i,j)∈IJ) ξij≧0 ((i,j)∈IJ); and wherein the modified optimization problem comprises: minimizew,b,η,ζ½∥w∥2+C·Σi=1mJ(i)·ηi+C·Σj=1nI(j)·ζj subject to yixijTw−yib+ηi+ζj≧1 ((i,j)∈IJ) ηi,ζj≧0. 4. The method of claim 3, further comprising: denoting a set P as attributes represented by pis; denoting a set Q as attributes represented by qjs; denoting a set U of attributes represented by uis; and denoting a set V of attributes represented by vjs, wherein the uis and the vjs are both common attributes of the T1 and T2; wherein J(i)=|{j:(i,j)∈IJ}|; wherein I(j)=|{i:(i,j)∈IJ}|; wherein I0={(i, 0):(∀j)(ui≠vj)}; and wherein IJ=I0∪{(i,j):ui=vj}. 5. The method of claim 4, wherein the table J is a new table based on a universal join of tables T1 and T2; and wherein the table J comprises |P|+|U|+|Q| columns; wherein class labels yi are associated with the rows of T1; wherein denote by z1, . . . , zl all the distinct values that appear as ui, such that for each k, k=1, . . . , l, denote Ik={i:ui=zk} and Jk={j:vi=zk}; wherein C is chosen as an arbitrary coefficient; and wherein b is a scalar. 6. The method of claim 5, wherein for each i, i=1, . . . , m, if there is no j such that uiT=vjT, then J has a row xi0T=(piT,uiT,0T), otherwise, J has rows of the form xijT=(piT, uiT, qjT) for every pair (i, j) such that uiT=vjT. 7. The method of claim 6, further comprising denoting by wP, wU and wQ projections of an unknown vector w on the sets P, U and Q, respectively. 8. The method of claim 1, further comprising solving the compact form by finding an optimal solution for: σk=mini∈Ik{yipiTwP−yib+ηi} and τkminj∈Jk{qjTwQ+ζj}. 9. The method of claim 1, further comprising developing a dual problem from the compact form of the modified optimization problem, the dual problem comprising: minimize ½Σi,i′yiyi′piTpi′αiαi′+½Σj,j′qjTqj′βjβj′+½Σk,k′zkTzk′γkγk′−Σi=1mγi subject to Σi=1myiαi=0 0≦αi≦CJ(i) (i=1, . . . , m) 0≦βi≦CI(j) (j=1, . . . , n) 0≦γk≦αi (k=1, . . . , l, i∈Ik) 0≦γk≦βj (k=1, . . . , l, j∈Jk). 10. The method of claim 9, further comprising solving the dual problem. 11. A computer program product for solving a support vector machine problem on table J, defined as the join of two tables T1 and T2, without explicitly joining the tables T1 and T2, wherein the table T1 has m rows (piT, uiT), i=1, . . . , m, and the table T2 has n rows (qjT, vjT), j=1, . . . , n, the computer program product comprising a computer readable storage medium having program instructions embodied therewith, the program instructions executable by computer to cause the computer to perform a method comprising: providing a primal optimization problem over a join of the tables T1 and T2; obtaining, by the computer, a modified optimization problem from the primal optimization problem; reducing penalty variables in the modified optimization problem by replacing the penalty variables in a form of ξij for each (i,j)∈IJ with the penalty variables in a form of ξij=ηi+ζj; obtaining a compact form of the modified optimization problem in which the compact form comprises the penalty variables in the form of ξijηiζj; and solving the compact form of the modified optimization problem. 12. The computer program product of claim 11, wherein the compact form comprises: Minimizew,b,η,ζ,σ,τ½∥wP∥2+½∥wU∥2½∥wQ∥2+C·Σi=1mJ(i)·ηi+C·Σj=1nI(j)·ζj, subject to yipiTwP−yib+ξi−σk≧0 (i∈Ik, k=1, . . . l) qjTwQ−τk≧0 (j∈Jk, k=1, . . . l) σk+zkTwU+τk≧1 (for k=1, . . . l such that Jk≠) σk+zkTwU≧1 (for k=1, . . . l such that Jk=) ξi≧0 (i=1, . . . , m); wherein the compact form includes auxiliary variables σ1, . . . , σl and τk for k=1, . . . l such that Jk≠. 13. The computer program product of claim 12, wherein the primal optimization problem comprises: minimizew,b,ξ½∥w∥2+C·Σ(i,j)∈IJξij subject to yixijTw−yib+ξij≧1 ((i,j)∈IJ) ξij≧0 ((i,j)∈IJ); and wherein the modified optimization problem comprises: minimizew,b,η,ζ½∥w∥2+C·Σi=1mJ(i)·ηi+C·Σj=1nI(j)·ζj subject to yixijTw−yib+ηi+ζj≧1 ((i,j)∈IJ) ηi,ζj≧0. 14. The computer program product of claim 13, further comprising: denoting a set P as attributes represented by pis; denoting a set Q as attributes represented by qjs; denoting a set U of attributes represented by uis; and denoting a set V of attributes represented by vjs, wherein the uis and the vjs are both common attributes of the T1 and T2; wherein J(i)=|{j:(i,j)∈IJ}|; wherein I(j)=|{i:(i,j)∈IJ}|; wherein I0={(i, 0):(∀j)(ui≠vj)}; and wherein IJ=I0∪{(i,j):ui=vj}. 15. The computer program product of claim 14, wherein the table J is a new table based on a universal join of tables T1 and T2; and wherein the table J comprises |P|+|U|+|Q| columns; wherein class labels yi are associated with the rows of T1; wherein denote by z1, . . . , zl all the distinct values that appear as ui, such that for each k, k=1, . . . , l, denote Ik={i:ui=zk} and Jk={j:vi=zk}; wherein C is chosen as an arbitrary coefficient; and wherein b is a scalar. 16. The computer program product of claim 15, wherein for each i, i=1, . . . , m, if there is no j such that uiT=vjT, then J has a row xi0T=(piT,uiT,0T), otherwise, J has rows of the form xijT=(piT, uiT, qjT) for every pair (i, j) such that uiT=vjT. 17. The computer program product of claim 16, further comprising denoting by wP, wU and wQ projections of an unknown vector w on the sets P, U and Q, respectively. 18. The computer program product of claim 11, further comprising solving the compact form by finding an optimal solution for: σk=mini∈Ik{yipiTwP−yib+ηi} and τk=minj∈Jk{qjTwQ+ζj}. 19. The computer program product of claim 11, further comprising developing a dual problem from the compact form of the modified optimization problem, the dual problem comprising: minimize ½Σi,i′yiyi′piTpi′αiαi′+½Σj,j′qjTqj′βjβj′+½Σk,k′zkTzk′γkγk′−Σi=1mγi subject to Σi=1myiαi=0 0≦αi≦CJ(i) (i=1, . . . , m) 0≦βi≦CI(j) (j=1, . . . , n) 0≦γk≦αi (k=1, . . . , l, i∈Ik) 0≦γk≦βj (k=1, . . . , l, j∈Jk). 20. The computer program product of claim 19, further comprising solving the dual problem.