Support vector machines are a super star in machine learning and data mining in the past decade. It reheats statistical learning in machine learning community. It is also one of the classifiers that work practically with applications to other disciplines such as bioinformatics.

In this post, I will give a brief review of the theory of this classifier and show an F# implementation using the quadratic programming (QP) solver in Microsoft Solver Foundation (MSF).

It may seem dull to start with mathematics. But the math in SVMs is actually quite intuitive and would be quite easy if one has some training in linear algebra and calculus. Also understanding the intuition behind the formulae is more important in the formulation itself. There are quite a lot of good SVM tutorials written by researchers. Thus I only write the important formulas that occur in the implementation. *The reader could link the F# code with formulas directly.*

My note is based on the machine learning course by Prof. Jaakkola [Jaakkola] and the Matlab implementation therein. The course material is highly comprehensive yet accurate.

## Background review for SVMs

Given a binary labeled dataset, where and is *1* or *-1*. Think about the data points plotted in a *d* dimensional (Euclidean) space, the linear SVM classifier is a hyperplane in the space and it “best” separates the two kinds of data points. Here “best” means that the hyperplane should have the largest margin, which is the distance from the plane to the sides of labeled points. Intuitively the larger the margin is, the more robust and confident the classifier is. As if the hyperplane shakes a little, it still classifies well for its being far from both sides.

Let’s take *d*=2 as the discussing example: data points are plotted in a 2-D plane. The aim is to draw a line to separate the two kinds of points such that the margin of the separation is maximized. In the following Figure, the line with maximum margin is shown. After some geometry work, the margin is calculated as , thus minimizing with the constraints that the two kinds of points are well separated () gives the max-margin hyperplane.

(Wikipedia picture)

Two important extensions to the above simplest SVM are:

1. **Allowing imperfect separation**, that is when a hyperplane could not separate the points perfectly, it is allowed to mis-separate some data points. This introduces a slack variable for each data point , when the point is on the correct side of the plane, the slack variable is 0, otherwise *it is measures the distance it goes from the plane minus 1, if the values goes to below zero set it zero*. (Please read the Note and reference section for more explanation.)

2. **Kernel**. The simplest SVM is a linear classifier in the original space. However, there are situations where linear classifier is not sufficient (even the best), one strategy is to do a non-linear transformation or mapping that maps a data point ** **to another space, so that a linear classifier in that space, which is non-linear in the original space, does a better separation of the data points. The kernel trick is that we don’t need to define explicitly; only the definition of inner product of that space is required: .

With the two extensions, the new maximum margin objective becomes:

subject to

The dual form:

subject to

The data points with its alpha value greater than 0 are called support vectors.

With the in the dual problem solved, the SVM classification hyperplane is recovered by:

The threshold parameter could be calculated by substituting back to the support vectors:

Any and (So that make sure data point is on the margin boundary, not in-between) would give a , to stabilize the solution, it is common practice to take the median of off the possible .

## The implementation

The solver is actually an Interior Point Solver, which could solve linear and quadratic programming with linear constraints.

Page 14-17 of MSF-SolverProgrammingPrimer.pdf show an example usage of the QP solver in Microsoft Solver Foundation. But the example is not complete and it does not demonstrate one very subtle part: how to set the coefficients for the quadratic terms.

The following code example shows a simple example with **the the comment regarding the coefficient setting**:

- #r @"C:\Program Files (x86)\Sho 2.0 for .NET 4\packages\Optimizer\Microsoft.Solver.Foundation.dll"
- open Microsoft.SolverFoundation.Common
- open Microsoft.SolverFoundation.Solvers
- open Microsoft.SolverFoundation.Services
- // test QP
- module TEST =
- (* minimize x^2 + y^2 + 3xy + 2x + y *)
- (* notice that in the InteriorPointSolver,
- the coefficients for xy & yx should be the same (so only set ONCE!)
- if we set both 3xy and 0yx, the solver takes the later coef.
- *)
- let solver = new InteriorPointSolver()
- let _, goal = solver.AddRow("dual objective value")
- solver.AddGoal(goal, 0, true)
- let _, x = solver.AddVariable("x")
- let _, y = solver.AddVariable("y")
- solver.SetCoefficient(goal, x, Rational.op_Implicit(2))
- solver.SetCoefficient(goal, y, Rational.op_Implicit(1))
- // for terms like x-y (where x != y), set its coef for only one time!
- solver.SetCoefficient(goal, Rational.op_Implicit(3), x, y)
- //solver.SetCoefficient(goal, Rational.Zero, y, x)
- solver.SetCoefficient(goal, Rational.op_Implicit(1), x, x)
- solver.SetCoefficient(goal, Rational.op_Implicit(1), y, y)
- let param = new InteriorPointSolverParams()
- solver.Solve(param) |> ignore
- //solver.Result
- let objectiveValue = solver.GetValue(0).ToDouble()
- let x0 = solver.GetValue(1).ToDouble()
- let y0 = solver.GetValue(2).ToDouble()
- x0*x0 + y0*y0 + 3.0 * x0 * y0 + 2. * x0 + y0
- //x0*x0 + y0*y0 + 0.0 * x0 * y0 + 2. * x0 + y0

The implementation of SVM is a straightforward translation of equations (1) to (5). The following shows the svm learning (**bulidSvm**) and testing (**svmClassify**) functions:

- open System
- type dataset =
- { features: float array array; // (instance = float array) array
- mutable labels: int array; //
- }
- with
- member x.NumSamples = x.features.Length
- module Array =
- let median (a:'a array) =
- let sorted = Array.sort a
- sorted.[sorted.Length / 2]
- module Kernel =
- let linear a b =
- Array.fold2 (fun acc p q -> acc + p * q) 0.0 a b
- let polynomial k a b =
- let dot = linear a b
- Math.Pow(1.0 + dot, k |> float)
- let gaussian beta a b =
- let diff = Array.fold2 (fun acc p q -> acc + (p-q)*(p-q)) 0.0 a b
- exp (-0.5 * beta * diff)
- module SVM =
- type svmmodel = {
- SVs:dataset;
- alpha:float array;
- kernel: float[] -> float[] -> float;
- w0:float;
- }
- with
- member x.NumSupporVectors = x.SVs.features.Length
- let buildSVM (ds:dataset) (C:float) (kernel:float[] -> float[] -> float) =
- let n = ds.features.Length
- let C = Rational.op_Implicit(C)
- let zero = Rational.Zero
- // create a interior point solver, which solves the QP problem
- let solver = new InteriorPointSolver()
- // set the objective value / goal
- let _, goal = solver.AddRow("dual objective value")
- // false == maximizing the objective value
- // the value of goal is (1)
- solver.AddGoal(goal, 0, false) |> ignore
- // add the Lagangian variables \alpha_i and set their bounds (0 <= \alpha_i <= C)
- let alpha = Array.create n 0
- for i=0 to n-1 do
- let _, out = solver.AddVariable("alpha_"+i.ToString())
- alpha.[i] <- out
- solver.SetBounds(out, zero, C)
- // add contraint: \sum_i \alpha_i * y_i = 0
- // equation (2)
- let _, sumConstraint = solver.AddRow("SumConstraint")
- solver.SetBounds(sumConstraint, zero, zero);
- for i=0 to n-1 do
- // set the coefs for the sum constraint
- // equation (2)
- solver.SetCoefficient(sumConstraint, alpha.[i], Rational.op_Implicit(ds.labels.[i]))
- // add the \alpha_i terms into the objective
- solver.SetCoefficient(goal, alpha.[i], Rational.One)
- // add the qudratic terms
- for j=0 to i do
- // coef = y_i * y_j * K(x_i, x_j)
- let coef = float(ds.labels.[i] * ds.labels.[j]) * (kernel ds.features.[i] ds.features.[j])
- if i=j then
- solver.SetCoefficient(goal, Rational.op_Implicit(-0.5 * coef), alpha.[i], alpha.[j])
- else
- solver.SetCoefficient(goal, Rational.op_Implicit(-coef), alpha.[i], alpha.[j])
- // use the default parameters
- let param = new InteriorPointSolverParams()
- solver.Solve(param) |> ignore
- // get the alpha values out
- let alphaValue = Array.init n (fun i -> solver.GetValue(i+1))
- (* print optimization result
- printfn "goal value = %A" (solver.GetValue(0).ToDouble())
- for i=1 to n do
- printfn "%A" (solver.GetValue(i).ToDouble())
- *)
- let alphaNew = new ResizeArray<Rational>()
- // extract the non-zero alpha values out and their corresponding support vectors
- let SVs =
- let feats = new ResizeArray<float[]>()
- let labs = new ResizeArray<int>()
- let maxAlpha = Array.max alphaValue
- let threshold = maxAlpha * Rational.op_Implicit(1e-8)
- for i=0 to n-1 do
- if alphaValue.[i] > threshold then
- feats.Add(ds.features.[i])
- labs.Add(ds.labels.[i])
- alphaNew.Add(alphaValue.[i])
- { features = feats |> Seq.toArray;
- labels = labs |> Seq.toArray;
- }
- // solve w_0 in the primal form
- let alphaNZ = alphaNew |> Seq.toArray
- // equation (5)
- let w0 =
- alphaNZ
- |> Array.mapi (fun i a ->
- if a = C then
- None
- else
- let mutable tmp = 0.0
- for j=0 to SVs.NumSamples-1 do
- tmp <- tmp + alphaNZ.[j].ToDouble() * (SVs.labels.[j] |> float) * (kernel SVs.features.[i] SVs.features.[j])
- Some ((float SVs.labels.[i]) - tmp)
- )
- |> Array.filter (fun v -> match v with None -> false | _ -> true)
- |> Array.map (fun v -> match v with Some v -> v | _ -> 0.0)
- |> Array.median
- // construct an svm record
- {
- SVs = SVs;
- alpha = alphaNZ |> Array.map (fun v -> v.ToDouble());
- kernel = kernel;
- w0 = w0;
- }
- let svmClassify (model:svmmodel) (ds:dataset) =
- // equation (4)
- let vals =
- ds.features
- |> Array.map (fun x ->
- let mutable sum = 0.0
- for i=0 to model.NumSupporVectors-1 do
- sum <- sum + model.alpha.[i] * (float model.SVs.labels.[i]) * (model.kernel model.SVs.features.[i] x)
- sum + model.w0
- )
- let nCorrect =
- Array.map2 (fun value label -> if (value > 0.0) && (label = 1) || (value < 0.0) && (label = -1) then 1 else 0) vals ds.labels
- |> Array.sum
- (float nCorrect) / (float ds.NumSamples), vals

Try on the iris data set we used in the Logistic Regression post:

let svm = buildSVM iris 10.0 Kernel.linear

let classifyResult = svmClassify svm iris

val svm : svm =

{SVs =

{features =

[|[|1.0; 6.2; 2.2; 4.5; 1.5|]; [|1.0; 5.9; 3.2; 4.8; 1.8|];

[|1.0; 6.3; 2.5; 4.9; 1.5|]; [|1.0; 6.7; 3.0; 5.0; 1.7|];

[|1.0; 6.0; 2.7; 5.1; 1.6|]; [|1.0; 5.4; 3.0; 4.5; 1.5|];

[|1.0; 4.9; 2.5; 4.5; 1.7|]; [|1.0; 6.0; 2.2; 5.0; 1.5|];

[|1.0; 6.3; 2.7; 4.9; 1.8|]; [|1.0; 6.2; 2.8; 4.8; 1.8|];

[|1.0; 6.1; 3.0; 4.9; 1.8|]; [|1.0; 6.3; 2.8; 5.1; 1.5|];

[|1.0; 6.0; 3.0; 4.8; 1.8|]|];

labels = [|-1; -1; -1; -1; -1; -1; 1; 1; 1; 1; 1; 1; 1|];};

alpha =

[|6.72796421; 10.0; 10.0; 10.0; 10.0; 6.475497298; 1.490719712;

8.547262902; 3.165478894; 10.0; 10.0; 10.0; 10.0|];

kernel = <fun:svm@226-32>;

w0 = -13.63716815;}

>

Real: 00:00:00.002, CPU: 00:00:00.000, GC gen0: 0, gen1: 0, gen2: 0

val classifyResult : float * float [] =

(0.97,

[|-2.787610618; -2.380530973; -1.42477876; -2.929203541; -1.681415929;

-1.96460177; -1.24778761; -6.106194692; -2.761061946; -2.973451328;

…. 4.203539824; 1.82300885|])

## Notes and references

The optimization in SVMs could be treated as a general QP problem as shown in our implementation. However, when the number of the data points is big, the QP grows too big to handle by the QP solver.

As a special QP problem, SVM has a lot of novel solutions proposed in the machine learning research area, e.g. the SMO approach [Platt] in LibSVM implementation and the Ball Vector Machine approach [Tsang] transforming the special QP into a (discrete) computational geometry problem, minimum enclosing ball. For practical usage of SVM, one usually uses these dedicated approaches.

Besides the maximum margin interpretation, SVMs also have theory roots in functional regularization theory, in which the optimization

is viewed as minimizing the error term while controlling the complexity of the function and *C* is a tradeoff parameter between these two. The reader is referred to the teaching notes and papers by Prof. Poggio [Poggio].

[Burges] A Tutorial on Support Vector Machines for Pattern Recognition. http://research.microsoft.com/en-us/um/people/cburges/papers/svmtutorial.pdf

[Jaakkola] http://courses.csail.mit.edu/6.867/.

[Platt] Sequential Minimal Optimization: A Fast Algorithm for Training Support Vector Machines. ftp://ftp.research.microsoft.com/pub/tr/tr-98-14.pdf

[Poggio] http://www.mit.edu/~9.520/spring10/.

[Tsang] http://www.cs.ust.hk/~ivor/cvm.html.

This comment has been removed by the author.

ReplyDeleteThe development of artificial intelligence (AI) has propelled more programming architects, information scientists, and different experts to investigate the plausibility of a vocation in machine learning. Notwithstanding, a few newcomers will in general spotlight a lot on hypothesis and insufficient on commonsense application. machine learning projects for final year In case you will succeed, you have to begin building machine learning projects in the near future.

DeleteProjects assist you with improving your applied ML skills rapidly while allowing you to investigate an intriguing point. Furthermore, you can include projects into your portfolio, making it simpler to get a vocation, discover cool profession openings, and Final Year Project Centers in Chennai even arrange a more significant compensation.

Data analytics is the study of dissecting crude data so as to make decisions about that data. Data analytics advances and procedures are generally utilized in business ventures to empower associations to settle on progressively Python Training in Chennai educated business choices. In the present worldwide commercial center, it isn't sufficient to assemble data and do the math; you should realize how to apply that data to genuine situations such that will affect conduct. In the program you will initially gain proficiency with the specialized skills, including R and Python dialects most usually utilized in data analytics programming and usage; Python Training in Chennai at that point center around the commonsense application, in view of genuine business issues in a scope of industry segments, for example, wellbeing, promoting and account.

The Nodejs Projects Angular Training covers a wide range of topics including Components, Angular Directives, Angular Services, Pipes, security fundamentals, Routing, and Angular programmability. The new Angular TRaining will lay the foundation you need to specialise in Single Page Application developer. Angular Training

This comment has been removed by the author.

ReplyDeleteGreat post Yin!

ReplyDeleteI was looking for info on how to use Microsoft Solver Foundation and I stumbled on your great blog again :)

"one _tragedy_ is to do a non-linear transformation" ? I think the spell correction got the better of your text ;-)

ReplyDelete"from the plane minus 1" - I hadn't seen that in my previous reading about SVMs. Isn't the slack just the distance past the boundary (without the -1)?

The SVM paper I was looking at was http://scribblethink.org/Work/Notes/svmtutorial.pdf, which on second reading does have the step that changes from a 0 to 1 slack excess.

ReplyDelete@philipoakley: Thanks for the correction on the spelling. "from the plane minus 1" -- please search "hinge loss function" to get more details of it:)

ReplyDeletesee this link:

ReplyDeletehttp://research.microsoft.com/en-us/um/people/manik/projects/trade-off/hinge.html

The blue line is the hinge loss, which is 0 when the difference is below 1.

This comment has been removed by the author.

ReplyDeleteMarch 2013

ReplyDeleteHi Yin.

Thanks for your blogs on F# MSF.

What is the latest status of F# and MSF?

Thanks, Art Scott

Awesome post, Call our Team member at Microsoft Online Support Phone Number Canada and at the other hand Microsoft Support Number Canada at 1-888-582-4887 for instant help and support if you have any issue with MS Window.

ReplyDeleteMicrosoft Office Support Canada 1-888-582-4887

Thanks for sharing this quality information with us. I really enjoyed reading.

ReplyDeleteマカフィー活性化, マカフィーインストール, マカフィーをインストールする, マカフィーセットアップ, マカフィーアンチウイルス, マカフィー総保護, マカフィーダウンロード, マカフィーは更新する, マカフィー問題, マカフィーをインストールできない, マカフィーのヘルプ

they are real people. Its not like they are computer parts or something. cursos de ti online

ReplyDeleterastgele görüntülü konuşma - kredi hesaplama - instagram video indir - instagram takipçi satın al - instagram takipçi satın al - tiktok takipçi satın al - instagram takipçi satın al - instagram beğeni satın al - instagram takipçi satın al - instagram takipçi satın al - instagram takipçi satın al - instagram takipçi satın al - binance güvenilir mi - binance güvenilir mi - binance güvenilir mi - binance güvenilir mi - instagram beğeni satın al - instagram beğeni satın al - polen filtresi - google haritalara yer ekleme - btcturk güvenilir mi - binance hesap açma - kuşadası kiralık villa - tiktok izlenme satın al - instagram takipçi satın al - sms onay - paribu sahibi - binance sahibi - btcturk sahibi - paribu ne zaman kuruldu - binance ne zaman kuruldu - btcturk ne zaman kuruldu - youtube izlenme satın al - torrent oyun - google haritalara yer ekleme - altyapısız internet - bedava internet - no deposit bonus forex - erkek spor ayakkabı - webturkey.net - karfiltre.com - tiktok jeton hilesi - tiktok beğeni satın al - microsoft word indir - misli indir

ReplyDeleteMua vé tại Aivivu, tham khảo

ReplyDeleteve may bay di my gia re

chuyến bay từ mỹ về việt nam tháng 1/2021

giá vé máy bay từ Toronto đến việt nam

có vé máy bay từ nhật về việt nam không

Có chuyến bay từ Hàn Quốc về Việt Nam không

Vé máy bay từ Đài Loan về Việt Nam

khách sạn cách ly ở việt nam

aşk kitapları

ReplyDeleteyoutube abone satın al

cami avizesi

cami avizeleri

avize cami

no deposit bonus forex 2021

takipçi satın al

takipçi satın al

takipçi satın al

takipcialdim.com/tiktok-takipci-satin-al/

instagram beğeni satın al

instagram beğeni satın al

btcturk

tiktok izlenme satın al

sms onay

youtube izlenme satın al

no deposit bonus forex 2021

tiktok jeton hilesi

tiktok beğeni satın al

binance

takipçi satın al

uc satın al

sms onay

sms onay

tiktok takipçi satın al

tiktok beğeni satın al

twitter takipçi satın al

trend topic satın al

youtube abone satın al

instagram beğeni satın al

tiktok beğeni satın al

twitter takipçi satın al

trend topic satın al

youtube abone satın al

takipcialdim.com/instagram-begeni-satin-al/

perde modelleri

instagram takipçi satın al

instagram takipçi satın al

takipçi satın al

instagram takipçi satın al

betboo

marsbahis

sultanbet

Tamamen Otomatik Sistem ile Siparişleriniz 7 Gün 24 Saat Hızlı ve Sorunsuz Bir Şekilde Tamamlanmaktadır. instagram takipçi satın al ve daha fazlası.

ReplyDeleteinstagram takipçi satın al

instagram beğeni satın al

instagram takipçi satın al

instagram takipçi satın al

instagram takipçi satın al

instagram takipçi satın al

instagram takipçi satın al

takipçi satın al

ucuz takipçi satın al

tiktok takipçi satın al

takipçi satın al

ReplyDeleteinstagram takipçi satın al

https://www.takipcikenti.com

marsbahis

ReplyDeletebetboo

sultanbet

marsbahis

betboo

sultanbet

www.escortsmate.com

ReplyDeleteescortsmate.com

https://www.escortsmate.com

anything thx all info post

ReplyDeleteVenüsbet Venüsbet Grandbetting Grandbetting Hiltonbet Betnano Marsbahis Meritking Asyabahis

thanks admin great article Hacklink

ReplyDeleteHacklink

Hacklink

Hacklink

Hacklink

Hacklink

Hacklink

SEO Fiyatları

Vezir Sosyal Medya

tiktok jeton hilesi

ReplyDeletetiktok jeton hilesi

referans kimliği nedir

gate güvenilir mi

tiktok jeton hilesi

paribu

btcturk

bitcoin nasıl alınır

yurtdışı kargo

Interesting Article. Hoping that you will continue posting an article having a useful information. Betnano giriş adresi

ReplyDeleteتست کرونا در منزل

ReplyDelete