replacing adverbs

Question

I am trying to rewrite the sentence below to replace the adverb 'far' (simply because I want to have as few adverbs as possible in my manuscript):

Specifically, their approach adds rejected applications with higher confidence, i.e. vectors far from the decision-hyperplane, iteratively to re-train the SVM.

But I cannot think of a reformulation for the sentence after 'i.e.'. Any suggestion?

UPDATE: Let me provide some background, as suggested, for the sentence.

• Field: Statistics/mathematics

• Vectors is a noun

• SVM is a mathematical model or algorithm. It stands for support vector machine

• iteratively to re-train the SVM: This means that the vectors are added one at a time, and after the vector is aded, the SVM algorithm runs. This recursion is repeated for all vectors.

• Reason to drop the word "far": I am working to improve my writting, especially avoiding passive voice. I have also seen that adverbs should be avoided, specifially adverbs with "ly" end.

Answer 1

First, far is not an adverb here. It's actually an adjective that's being used as part of a prepositional phrase.

However, the problem with your sentence doesn't start after i.e., it starts before it.

Specifically, you are leaving a very large gap between the adverb iteratively and the verb to add that it is modifying.

A possible rephrasing would would look like this:

Specifically, their approach iteratively adds rejected applications with higher confidence, i.e. vectors far from the decision-hyperplane, to retrain the SVM.

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This should fix the main problem, which is the misplaced adverb.

If you find that the parenthetical nonessential information is still detracting from the flow of the sentence, you can use actual parentheses to help with the parsing:

Specifically, their approach iteratively adds rejected applications with higher confidence (i.e. vectors far from the decision-hyperplane) to retrain the SVM.

Or you can break it up into two sentences:

Specifically, their approach iteratively adds rejected applications with higher confidence to retrain the SVM. Note that rejected applications with higher confidence is another way of referring to vectors far from the decision-hyperplane.

Answer 2

It sounds as if you are describing a procedure for modifying the separating hyperplane in a support vector machine (SVM).

The problem is that vector has two meanings in this context, and this is creating confusion in your sentence.

The input to an SVM of the type you are describing is a series of classified observations, where each observation is a point in an n-dimensional space, and the classifier is a binary 1 or 0. The observation point is described by its observation vector.

The hyperplane is described by a linear equation with n-1 coefficients. These coefficients are calculated so that the hyperplane separates the “0-classified” points from the “1-classified” points as much as possible.

The meaning of as much as possible is defined in terms of the support vectors, which are the normal vectors from the observation points to the hyperplane. Basically, “best” hyperplane coefficients are those which minimize the sum of the squared lengths of the normal vectors.

In the process you are describing, when a new observation point arrives, and is found to lie far from the existing hyperplane, this new point is added to the set of points used to compute the hyperplane coefficients, and new coefficients are calculated. This may be loosely referred to as “iteratively retraining the SVM”.

The support vector for the newly added point may be long, but it is not technically correct to say that this vector is far from the hyperplane, since it actually connects the hyperplane to the observation point.

Nor is it accurate to say that the observation vector is far from the hyperplane. The observation vector connects the origin point in the n-dimensional space with the observation point. It is not impossible for the hyperplane to include the origin, and also not impossible for the origin and the observation points to lie on opposite sides of the hyperplane.

Using the mathematical concepts more carefully, your description could be rephrased as follows:

Their approach allows for the retraining of the SVM after each new observation point. If a point is classified with high confidence (lying beyond a specified distance from the separating hyperplane), then the hyperplane coefficients are updated. This is appled only when the classification corresponds to a rejection.