Consider a labelled data set having d attributes and n points in two linearly separable classes. In which of the following situations will the classes necessarily remain linearly separable? [Select ALL that apply.]

Question

BlackTom AI · Accepted Answer

The question asks about how certain changes to a labelled, linearly separable dataset of n points in d attributes affect linear separability, and to select all changes that necessarily preserve separability.

Option 1: An attribute is removed.
- When you remove an attribute (dimension), you project the data into a lower-dimensional space. It is possible for a set that is linearly separable in the original d dimensions to become non-separable after projection, because the removed feature might have carried the discriminative information needed to separate the classes. Therefore, separability is not guaranteed to be preserved, and this option does not necessarily preserve linear separability.

Option 2: A new labelled point is added.
- Adding a single new point with a label can disrupt separability if that point cannot be placed on the correct side of any existing separating hyperplane. Since the dataset must be strictly separable by a hyperplane across all points, inserting a new point with a conflicting label can destroy separability. Thus, this option does not necessarily preserve linear separability.

Option 3: A labelled point is removed (assuming both classes still have points left after the removal).
- Removing a point cannot create a new obstruction to separability; if there exists a separating hyperplane for the full dataset, removing a point leaves that hyperplane still valid for the remaining points (or there could exist other hyperplanes as well). Therefore, linear separability is preserved or remains possible after removal, so this option does preserve separability.

Option 4: A new attribute is added.
- Adding a new attribute increases the dimensionality of the feature space. Importantly, you can keep the original separating hyperplane by setting the weight for the new attribute to zero, effectively ignoring the new feature and using the same decision boundary as before. Hence, the dataset remains linearly separable in the augmented space, so this option also preserves separability.

In summary, the changes that necessarily preserve linear separability are those that do not disrupt the existing discriminative structure or provide a way to ignore the new dimension. These are the cases of removing a point and adding a new attribute. Conversely, removing an attribute or adding a new labelled point can jeopardize separability, so they are not guaranteed to preserve it.

Consider a labelled data set having d attributes and n points in two linearly separable classes. In which of the following situations will the classes necessarily remain linearly separable? [Select ALL that apply.]多项选择题

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