A metric learning perspective of SVM: on the relation of SVM and LMNN

Support Vector Machines, SVMs, and the Large Margin Nearest Neighbor algorithm, LMNN, are two very popular learning algorithms with quite different learning biases. In this paper we bring them into a unified view and show that they have a much stronger relation than what is commonly thought. We analyze SVMs from a metric learning perspective and cast them as a metric learning problem, a view which helps us uncover the relations of the two algorithms. We show that LMNN can be seen as learning a set of local SVM-like models in a quadratic space. Along the way and inspired by the metric-based interpretation of SVM s we derive a novel variant of SVMs, epsilon-SVM, to which LMNN is even more similar. We give a unified view of LMNN and the different SVM variants. Finally we provide some preliminary experiments on a number of benchmark datasets in which show that epsilon-SVM compares favorably both with respect to LMNN and SVM.


Introduction
Support Vector Machines, [2], and metric learning algorithms, [15,8,9,16], are two very popular learning paradigms with quite distinct learning biases. In this paper we focus on SVMs and LMNN, one of the most prominent metric learning algorithms, [15]; we bring them into a unified view and show that they have a much stronger relation than what is commonly accepted. We show that SVM can be formulated as a metric learning problem, a fact which provides new insights to it. Based on these insights and employing learning biases typically used in metric learning we propose -SVM, a novel SVM variant which is shown to empirically outperform SVM. -SVM relates, but is simpler, to the radius-margin ratio error bound optimization that is often used when SVM is coupled with feature selection and weighting, [12,5], and multiple kernel learning [1,4,7]. More importantly we demonstrate a very strong and previously unknown connection between LMNN and SVM. Until now LMNN has been considered as the distance-based counterpart of SVM, in the somehow shallow sense that both use some concept of margin, even though their respective margin concepts are defined differently, and the hinge loss function, within a convex optimization problem. We show that the relation between LMNN and SVM is much deeper and demonstrate that LMNN can be seen as a set of local SVM-like classifiers in a quadratic space. In fact we show that LMNN is even more similar to -SVM than SVM. This strong connection has the potential to lead to more efficient LMNN implementations, especially for large scale problems and vice versa, to lead to more efficient schema of multiclass SVM. Moreover the result is also valid for other large margin metric learning algorithms. Finally we use the metric-based view to provide a unified view of SVM, -SVM and LMNN.
Overall the main contribution of the paper is the unified view of LMNN, SVM and the variants of the latter. Along the way we also devise a new algorithm, -SVM, that combines ideas from both SVM and LMNN and finds its support in the SVM error bound.
In the next section we will briefly describe the basic concepts of SVM and LMNN. In section 2 we describe SVM in the metric learning view and the new insights that this view brings. We also discuss some invariant properties of SVM and how metric learning may or may not help to improve SVM. Section 3 describes -arXiv:1201.4714v1 [cs.LG] 23 Jan 2012 SVM, a metric-learning and SVM inspired algorithm. Section 4 discusses the relation of LMNN and SVM, and provides a unified view of the different SVM and LMNN variants. In section 6 we give some experimental results for -SVM and compare it with SVM and LMNN. Finally we conclude in section 7.

Basic SVM and LMNN concepts
We consider a binary classification problem in which we are given a set of n learning instances S = {(x 1 , y 1 ), ..., (x n , y n )}, x i ∈ R d , where y i is the class label of the x i instance and y i ∈ {+1, −1}. We denote by x p the l p norm of the x vector. Let H w be a hyperplane given by SVMs learn a hyperplane H w which separates the two classes and has maximum margin γ, which is, informally, the distance of the nearest instances from H w : [2]. The optimization problem that SVMs solve is: which is usually rewritten to avoid the uniform scaling problem with w, b, as: The margin maximization is motivated by the SVM error bound which is a function of the R 2 /γ 2 ratio; R is the radius of the smallest sphere that contains the learning instances. Standard SVMs only focus on the margin and ignore the radius because for a given feature space this is fixed. However the R 2 /γ 2 ratio has been used as an optimization criterion in several cases, for feature selection, feature weighting and multiple kernel learning [1,12,4,5,7]. The biggest challenge when using the radius-margin bound is that it leads to non convex optimization problems.
In metric learning we learn a Mahalanobis metric parametrized by a Positive Semi-Definite (PSD) matrix M under some cost function and some constraints on the Mahalanobis distances of same and different class instances. The squared Mahalanobis distance has the following form d 2 . M can be rewritten as M = L T L, i.e the Mahalanobis distance computation can be considered as a two-step procedure, that first computes a linear transformation of the instances given by the matrix L and then the Euclidean distance in the transformed space, i.e LMNN, one of the most popular metric learning methods, also works based on the concept of margin which is nevertheless different from that of SVM. While the SVM margin is defined globally with respect to a hyperplane, the LMNN margin is defined locally with respect to center points. The LMNN margin, γ x0 , of a center point, instance x 0 , is given by: where targ(x 0 ) is the set of the target neighbors of x 0 , which is defined as targ( The neighborhood can be defined either as a k nearest neighbors or as a sphere of some radius.
LMNN maximizes the sum of the margins of all instances. The underlying idea is that it learns a metric M or a linear transformation L which brings instances close to their same class center point while it moves away from it different class instances with a margin of one. This learning bias is commonly referred as large margin metric learning [13,15,14,10]. LMNN optimization problem is [15]:

SVM under a metric learning view
We can formulate the SVM learning problem as a metric learning problem in which the learned transformation matrix is diagonal. To do so we will first define a new metric learning problem and then we will show its equivalence to SVM.
We start by introducing the fixed hyperplane H 1 , the normal vector of which is 1 (a d-dimensional vector of ones), i.e. H 1 : We now define the margin γ of two classes with respect to the hyperplane H 1 as the minimum absolute difference of the distances of any two, differentclass instances,x i ,x j , projected to the norm vector of H 1 hyperplane, which can be finally written as: γ = min i,j,yi =yj In fact this is the SVM margin, see equation (17) in Appendix.
We are interested in that linear transformation, diag(W) = w, for which it holds y i (w T x i + b) ≥ 0, i.e. the instances of the two classes lie on two different sides of the hyperplane H 1 , and which has a maximum margin. This is achieved by the following metric-learning optimization problem 1 : Additionally we prefer the W transformation that places the two classes "symmetrically" with respect to the H 1 hyperplane, i.e. the separating hyperplane is at the middle of the margin instances of the two classes: Notice that this can always be achieved by adjusting the translation parameter b by adding to it the γ1−γ2 2 value, where γ 1 and γ 2 are respectively the margins of the y 1 and y 2 classes to the H 1 hyperplane. Hence the parameter b of the H 1 hyperplane can be removed and replaced by an equivalent translation transformation b. From now on, we assume H 1 : 1 T x + 0 = 0 and we add a translation b after the linear transformation W. With this 'symmetry' preference, (6) can be reformulated as (see the detailed proof in Appendix): This optimization problem learns a diagonal transformation W and a translation b which maximize the margin and place the classes 'symmetrically' to the H 1 : 1 T x = 0 hyperplane. However this optimization problem scales with the w (as is the case with the SVM formulation (1)). Therefore we need a way to control the uniform scaling of w. One way is to fix some norm of w e.g. w p = 1, and the optimization problem becomes: Notice that both problems (7) and (8) are still different from the standard SVM formulation given in (1) or (2). However it can be shown that they are in fact equivalent to SVM; for the detailed proof see in the Appendix.
Thus we see that starting from a Mahalanobis metric learning problem (5), i.e learning a linear transformation W, and with the appropriate cost function and constraints on pairwise distances, we arrive to a standard SVM learning problem. We can describe SVM in the metric learning jargon as follows: it learns a diagonal linear transformation W and a translation b which maximize the margin and place the two classes symmetrically in the opposite sides of the hyperplane H 1 : 1 T x = 0. In the standard view of SVM, the space is fixed and the hyperplane is moved around to achieve the optimal margin. In the metric view of SVM, the hyperplane is fixed to H 1 and the space is scaled, W, and then translated, b, so that the instances are placed optimally around H 1 . Introducing a fixed hyperplane H 1 will provide various advantages in relation to the different radius-margin SVM versions and LMNN as we will show soon. It is also worth to note that the SVM metric view holds true for any kernel space, since its final optimization problem is the same as that of standard SVM, i.e we can kernelize it directly as SVM.
From the metric learning perspective, one may think that learning a full linear transformation instead of a diagonal could bring more advantage, however as we will see right away this is not true for the case of SVM. For any linear transformation L (full matrix), the distance of Lx to the hyperplane where D L is a diagonal matrix, in which the k diagonal element corresponds to the sum of the elements of the kth column of L: D Lkk = i L ik . So for any full transformation matrix L there exists a diagonal transformation D L which has the same signed distance to the hyperplane. This is also true for any hyperplane w T x + b where w does not contain any zero elements. Thus learning a full matrix does not bring any additional advantage to SVM.

3
-SVM, an alternative to the radius-margin approach This bias is often implemented through local or global constraints on the pair-wise distances of sameclass and different-class instances which make the between class distance large and the within class distance small. Under the metric view of SVM we see that the learned linear transformation does control the between class distance by maximizing the margin, but it ignores the within class distance. Inspired by the metric learning biases we will extend the SVM under the metric view and incorporate constraints on the within-class distance which we will minimize.
We can quantify the within class distance with a number of different ways such as the sum of instance distances from the centers of their class, as in Fisher Discriminant Analysis (FDA) [11], or by the pairwise distances of the instances of each class, as it is done in several metric learning algorithms [16,3,15]. Yet another way to indirectly minimize the within class distance is by minimizing the total data spread while maximizing the between class distance. One measure of the total data spread is the radius, R, of the sphere that contains the data; so by minimizing the radius-margin ratio, R 2 /γ 2 , we can naturally minimize the within class distance while maximizing the between. Interestingly the SVM theoretical error bound points towards the minimization of the same quantity, i.e. the radiusmargin ratio, thus the proposed metric learning bias finds its theoretical support in the SVM error bound. However optimizing over the margin and the radius poses several difficulties since the radius is computed in a complex form, [12,4,7].
We can avoid the problems that are caused by the radius by using a different quantity to indirectly control the within class distance. We propose to minimize instead of the radius, the sum of the instance distances from the margin hyperplane, this sum is yet another way to control the data spread. We will call the resulting algorithm, which in addition to the margin maximization also minimizes the within-class-distance through the sum of the instance distances from the margin hyperplane, -SVM. The minimization of the sum has a similar effect to the minimization of the SVM radius, Figure 1. In a later section we will show that -SVM can be seen as a bridge between LMNN and SVM.
We define the optimization problem of -SVM as follows: select the transformations, one linear diagonal, diag(W) = w, and one translation, b, which maximize the margin and keep the instances symmetrically and within a small distance from the H 1 hyperplane. This is achieved by the following optimization problem: penalizes instances which lie on the correct side of the hyperplane but are far from the margin. max(0, 1 − y i (w T x i + b) is the SVM hinge loss which penalize instances violating the margin. (9) is equivalent to: where ξ i is the standard SVM slack variable for instance i and η i the distance of that instance from the margin hyperplane. The dual form of this optimization problem is: (11) Note that we have two hyper-parameters C and λ, typically we will assign a higher value to the C since we tolerate less the violations of the margin compared to a larger data spread. The two parameters control the trade off between the importance of the margin and the within data spread.

On the relation of LMNN and SVM
In this section we demonstrate the relation between LMNN and SVM. Lets define a quadratic mapping . We will show that the LMNN margin in the original space is the SVM margin in this quadratic space.
Concretely, the squared distance of an instance x from a center point instance x l in the linearly transformed space that corresponds to the Mahalanobis distance M learned by LMNN can be expressed in a linear form of Φ(x). Let L be the linear transformation associated with the learned Mahalanobis distance M. We have: A more general formulation of the LMNN optimization problem (4) which reflects the same learning bias of LMNN, i.e for each center point, keeps the same-class instances close to the center and pushes different-class instances outside the margin, is: Standard LMNN puts one extra constraint on each γ x l , it sets each of them to one. With this constraint problem (13) reduces to (4). We can rewrite problem (13) as: In this formulation, visualized in fig. 2(a), the target neighbors (marked with ×) of the x l center point are constrained within the C l circle with radius R l and center Lx l while the instances that have a label which is different from that of x l (marked with ) are placed outside the circle C l with center also Lx l and radius R l = R 2 l + γ x l . We denote by β l the difference R l − R l of the radii of the two circles.
If we replace all the d 2 (Lx, Lz) terms in problem (14) with their linear equivalent in the quadratic space, problem (12), and break up the optimization problem to a series of n optimization problems one for each instance x l then we get for each x l the following optimization problem: + λ where w l is not an independent variable but its quadratic and linear components are related as described above (formula (12)).
The instance set B(x l ) = {x i ∈ targ(x l )} ∪ {x i |y i = y l } is the set of all target neighbors and different class instances. In the formula (15) we replace b l by −w T l Φ(x l ) since w T l Φ(x l ) + b l = 0, and b l = b l − (R 2 l + γ x l /2). We denote the hyperplane w T l Φ(x) + b l = 0 by H l , as in fig. 3. Now let y li := y l y i , so y li = 1, ∀x i ∈ targ(x l ) and y lj = −1, ∀(x j |y i = y l ). Therefore −y li (w T l Φ(x i ) + b l ) ≥ γ x l /2 − ξ li , ∀x i ∈ B(x l ). Without loss of generality we can assume that y l = −1 and problem (15) becomes: + λ where w l is still constrained as in (15). It is worth to note that the second constraint of (16) will ensure that d 2 M (x i , x l ) is bigger than or equal to 0, i.e it will almost ensure the PSD of the matrix M 2 . However due to the specific structure of w l , i.e. quadratic and linear part, the PSD constraint of M is guaranteed for any x.
We can think of problem (16) as an extension of an SVM optimization problem. Its training set is the B(x l ) set, i.e. the target neighbors of x l and all instances with different class label from x l . Its cost function includes, in addition to the term that maximizes the margin, also a sum term which forces the target neighbors of x l to have small w T l Φ(x i ) + b l values, i.e. be at small distance from the H l hyperplane.
Minimizing the target neighbor distances from the H l hyperplane makes the distance between support vectors and H l small. It therefore has the effect of bringing the negative margin hyperplane of H l close to H l , bringing thus also the target neighbors close to the negative margin hyperplane. In other words the optimization problem favors a small width for the band that is defined by H l and the negative margin hyperplane described above which contains the target neighbors.
There is an even closer relation of LMNN with a local -SVM applied in the quadratic space that we will describe by comparing the optimization problems given in (9) and (16). -SVM has almost the same learning bias as LMNN, the former maximizes the margin and brings all instances close to the margin hyperplanes, the latter also maximizes the margin but only brings the target neighbors close to the margin hyperplane. For each center point x l the LMNN formulation, problem (16), is very similar to that of -SVM, problem (9), applied in the quadratic space. The second term of the cost function of -SVM together with its constraints force all instances to be close to their respective margin hyperplane, this can be seen more clear in the minimization of the η i in problem (10). In LMNN, problem (16), the third term of the cost function plays a similar role by forcing only the target neighbors to be close to the H l hyperplane. This in turn has the effect, as mentioned above, to reduce the width of the band containing them and bringing them closer to their margin hyperplane; unlike -SVM no such constraint is imposed on the remaining instances. So LMNN is equivalent to n local, modified, -SVMs, one for each instance. -SVM controls the within class distance using the distance to the margin hyperplane and LMNN using the distance to the H l hyperplane.
Note that the n optimization problems are not independent. The different solutions w l have a common component which corresponds to the d(d + 1)/2 quadratic features and is given by the w quad vector. The linear component of w l , w lin l , is a function of the specific x l , as described above. Overall learning a transformation L with LMNN, eq. (4), is equivalent to learning a local SVM-like model, given by H l , for each x l center point in the quadratic space according to problems (15,16). Remember that the w l , b l of the different center points are related, i.e. their common quadratic part is w quad . If we constrain w l , b l to be the same for all x l and drop the PSD constraints on w l then we get the (global) -SVM-like solution.
Visualization: In fig. 2(b) we give a visualization of problem (15) in the quadratic space Φ(x). Figure 2(b) gives the equivalent linear perspective in the quadratic space of the LMNN model in the original space given in fig. 2(a). The center Lx l of the C l circle in fig. 2(a) corresponds to the H l hyperplane in fig. 2(b); the C l circle with center Lx l and radius R l = R l + β l /2 corresponds to the H l separating hyperplane in fig. 2(b). Figure 3(a) illustrates the different local linear models in the quadratic space. We can combine these different local models by employing the metric learning view of SVM and make the relation of LMNN and SVM even more clear. Instead of having many local SVM-like hyperplanes we bring each point Φ(x l ) around the H 1 : 1x + 0 = 0 hyperplane by applying to it first a W l diagonal transformation, W l = w l , and then a translation ( fig. 3(b)). As before with w l the different W l transformations have a common component, which corresponds to the first d(d + 1)/2 elements of the diagonal associated with the quadratic features, given by the w quad vector, and an instance dependent component that corresponds to the linear features which is given by w lin l = −2L T Lx l ; thus the translation transformation is also a function of the spe-cific point x l . Notice that the common component has many more elements than the instance specific component. There is an analogy to multitask learning where models are learned over different tasks-datasets are forced to have parts of their models the same.   Prediction phase Typically after learning a metric through LMNN a k-NN algorithm is used to predict the class of a new instance. Under the interpretation of LMNN as a set of local -SVM linear classifiers this is equivalent to choosing the k local hyperplanes H l : w T l Φ(x) + b l = 0 which are farther away from the new instance and which leave both the new instance and their respective center points on the same side. The farther away the new instance is from an H l local hyperplane the closer it is to the center point associated to H l . In effect this means that we simply choose those hyperplanes-classifiers which are more confident on the label prediction of the new instance, and then we have them vote. In a sense this is similar to an ensemble learning schema in which each time we want to classify a new instance, we select those classifiers that are most confident. Then we have them vote in order to determine the final prediction.

A unified view of LMNN, SVM and its variants
The standard SVM learning problem only optimizes the margin. On the other hand, LMNN, as well as the different variants of radius-margin based SVM under the metric learning view (briefly mentioned in section 3), and -SVM, have additional regularizers that control directly or indirectly the within class distance. LMNN is much closer to a collection of local -SVMs than to local SVMs, since both LMNN and -SVM have that additional regularizer of the within class distance. We briefly summarize the different methods that we discussed: • In standard SVM the space is fixed and the separating hyperplane is moved to find an optimal position in which the margin is maximized. • In standard LMNN we find the linear transformation that maximizes the sum of the local margins of all instances while keeping the neighboring instances of the same class close. • In the metric learning view of SVM we find a linear diagonal transformation W, diag(W) = w plus a translation b that maximize the margin to the fixed hyperplane H 1 , clamping the norm of w to 1, or maximize the norm of w while clamping the margin to 1. • In -SVM we find a linear diagonal transformation and a translation that maximize the margin with respect to the fixed hyperplane H 1 and keep all instances close to H 1 . • Finally in the interpretation of LMNN as a collection of local SVM-like classifiers we find for each instance x a linear transformation and a translation that clamp the margin to one and keep the target neighbors of x within a band the width of which is minimized. These transformations have common components as described in the previous section. LMNN is set of local related -SVMs in the quadratic space.
We will now give a high level picture of the relations of the different methods in the quadratic space Φ(x) by showing how from one method we can move to another by adding or dropping constraints; the complete set of relations is given in Figure 4.
We start with n local non-related SVMs, if we add a constraint on the within-class distance to each of them we get n non-related -SVMs. If we add additional constrains that relate the different local -SVMs, namely by constraining their quadratic components w quad to be the same and PSD, and their linear component w lin l to be a function of w quad and the center point x l then we get LMNN.
If we go back to the original set of non-related SVMs and add a constraint that forces all of them to have the same w l and b l we get the standard global SVM.
Similarly if in the collection of the local -SVMs we add a constraint that forces all of them to have the same optimal hyperplane then we also get the global -SVM. In that case all the component classifiers of the ensemble reduce to one classifier and no voting is necessary.
LMNN constrains the n local -SVMs to have the same quadratic component, w quad , makes the linear components dependent on w quad and their respective center points, accommodating like that the local information, and constrains the quadratic components w quad to reflect the L T L PSD matrix, equation (12). On the other hand both the global SVM and -SVM also constrain their quadratic components to be the same but remove the constraint on the PSD property of w quad ; both constrain all their linear components to be the same, thus they do not incorporate local information. The last constraint is much more restrictive than the additional constrains of LMNN, as a result the global SVM and -SVM models are much more constrained than that of LMNN. Although we demonstrate the relations only for the case of SVM and LMNN they also hold for all other large margin based metric learning algorithms [13,15,14,10].
-SVM builds upon two popular learning biases, namely large margin learning and metric learning and exploits the strengths of both. In addition it finds theoretical support in the radius-margin SVM error bound. -SVM can be seen as a bridge which connects SVM and existing large margin based metric learning algorithms, such as LMNN.

Experiments and results
In addition to studying in detail the relation between the different variants of SVM and metric learning we also examine the performance of the -SVM algorithm.
We experiment with -SVM, eq. (10), and compare its performance to that of a standard l 1 soft margin SVM, and LMNN, with the following kernels: linear, polynomial with degree 2 and 3, Gaussian with σ = 1. C is chosen with 10-fold inner Cross-Validation from the set {0.1, 1, 10, 100, 1000}. For -SVM we choose to set λ  to C/3 reflecting the fact that we tolerate less the margin violations than a larger distance form the margin. We used ten datasets mainly from the UCI repository [6]. Attributes are standardized to have zero mean and one variance; kernels are normalized to have a trace of one. The number of target neighbors for LMNN is set to 5 and its γ parameter is set to 0.5, following the default settings suggested in [15]. We estimated the classification error using 10-fold CV. The results are given in Table 1. Overall each algorithm is applied 40 times (four kernels × ten datasets). Comparing -SVM with SVM we see that the former has a lower error than SVM in 19 out of the 40 applications and a higher error in only nine. A similar picture appears in the -SVM, LMNN, pair where the former has a lower error in 20 out of the 40 applications and a higher in only eight. If we break down the comparison per kernel type, we also see that, i.e. -SVM has a systematic advantage over the other two algorithms no matter which kernel we use.
To examine the statistical significance of the above results we use the McNemar's test of significance, with a significance level of 0.05. Comparing algorithms A and B on a fixed dataset and a fixed kernel algorithm A is credited with one point if it was significantly better than algorithm B, 0.5 points if there was no significance difference between the two algorithms, and zero points otherwise. In Table 2 we report the overall points that each algorithms got and in addition we break them down over the different kernels. -SVM has the highest score with 46 points followed by SVM with 37.5 and LMNN with 36.5. The advantage of -SVM is much more pronounced in the case of the Gaussian kernel. This could be attributed to its additional regularization on the within-class distance which makes it more appropriate for very high dimensional spaces.

Conclusion
In this paper, we have shown how SVM learning can be reformulated as a metric learning problem. Inspired by this reformulation and the metric learning biases, we proposed -SVM, a new SVM-based algorithm in which, in addition to the standard SVM constraints we also minimize a measure of the within class distance. More importantly the metric learning view of SVM helped us uncover a so far unknown connection between the two seemingly very different learning paradigms of SVM and LMNN. LMNN can be seen as a set of local SVM-like classifiers in a quadratic space, and more precisely, as a set of local -SVM-like classifiers. Finally preliminary results show a superior performance of -SVM compared to both SVM and LMNN. Although our discussion was limited to binary classification, it can be extended to the multiclass case. Building on the SVM-LMNN relation, our current work focuses on the full analysis of the multiclass case, a new schema for multiclass SVM which exploits the advantages of both LMNN and kNN in multiclass problems, and finally the exploration of the learning models which are in between the SVM, LMNN models.
With the 'symmetry' preference as described after problem (6), the second constraint of problem (6) can be replaced by the constraint: y i (w T x i + b) ≥ γ/2. Moreover, we can always replace the squared values in problem (6)  If the constraint y i (w T x i + b) ≥ γ/2 is satisfied then w T x i + b and y i have the same sign; therefore any two instances ∀x i , x j which have different labels, (y i = y j ), will lie on the opposite sides of the hyperplane. Hence: Therefore the first constraint of problem (6) is always satisfied if the constraint y i (w T x i + b) ≥ γ/2 is satisfied, thus (6) is equivalent to (7).

Equivalence of (8) to standard SVM formulation
We will show that problem (8) is equivalent to standard SVM. In fact (1) also scales with uniform scaling of w, b and to avoid this problem, w γ is fixed to 1 which lead to the second formula (2). We will show that the two ways of avoiding scaling problem, by forcing w γ = 1 or by forcing w = 1 are equivalent.
Indeed, another way to avoid the scaling problem is to find a quantity which is invariant to the scaling of w. Let γ = t w p , hence t : 0 → ∞, and lets fix w p = 1. Then let P be the feasible set of w which satisfies ( w p = 1 and y i (w T x i + b) ≥ 0, ∀i), and Q = {w| w p = 1 and y i (w T x i + b) ≥ t w p ), ∀i} Notice that, if t = 0 then Q ≡ P , if t > 0 then Q ⊆ P , and if t > t max then Q will be empty. For another value of w p , w p = λ, the corresponding feasible sets are P λ and Q λ . There is a one to one mapping from P to P λ , and from Q to Q λ , and the t max λ which makes Q λ empty is the same as t max . So t max is invariant to the scaling of w. Therefore (8) is equivalent to: The value of the geometric margin here is fixed to 1/ √ d while in standard SVM the geometric margin is γ = 1/ w 2 2 . Using the l 2 norm of w, we get a formulation which is equivalent to that of the hard margin SVM given in (2). Using the l 1 norm of w, we get the 1-norm SVM [17].