Is PCA convex?
No, the usual formulations of PCA are not convex problems.
Are Least Squares convex?
Linear least squares problems are convex and have a closed-form solution that is unique, provided that the number of data points used for fitting equals or exceeds the number of unknown parameters, except in special degenerate situations.
How do you prove a loss function is convex?
Is our loss function convex? A function f : Rd → R is convex if for all a,b ∈ Rd and 0 <θ< 1, f (θa + (1 − θ)b) ≤ θf (a) + (1 − θ)f (b). A function f : R → R is convex if its second derivative is ≥ 0 everywhere.
Is linear regression convex?
The Least Squares cost function for linear regression is always convex regardless of the input dataset, hence we can easily apply first or second order methods to minimize it.
Is PCA optimal?
PCA is often used in this manner for dimensionality reduction. PCA has the distinction of being the optimal orthogonal transformation for keeping the subspace that has largest “variance” (as defined above).
What is strongly convex?
Intuitively speaking, strong convexity means that there exists a quadratic lower bound on the growth of the function. This directly implies that a strong convex function is strictly convex since the quadratic lower bound growth is of course strictly grater than the linear growth.
What is the difference between convex and strictly convex?
Geometrically, convexity means that the line segment between two points on the graph of f lies on or above the graph itself. See Figure 2 for a visual. Strict convexity means that the line segment lies strictly above the graph of f, except at the segment endpoints.
Why the loss function is convex?
If the function grows faster then linear with deviation from the minimum, then the function is convex. For multidimensional case for any projection on a plane, cutting the loss hypersurface you need to have the same picture as for 1-dimensional case.
Is the loss function always convex?
Fortunately, hinge loss, logistic loss and square loss are all convex functions.
Why is MSE convex?
Answer in short: MSE is convex on its input and parameters by itself. But on an arbitrary neural network it is not always convex due to the presence of non-linearities in the form of activation functions.
Is logistic regression convex?
4.5. The method most commonly used for logistic regression is gradient descent. Gradient descent requires convex cost functions. Mean Squared Error, commonly used for linear regression models, isn’t convex for logistic regression. This is because the logistic function isn’t always convex.
What is threshold in PCA?
SUMMARY. In functional principal component analysis (fPCA) a threshold is chosen to define the number of retained principal components, which corresponds to the amount of preserved information. A variety of thresholds have been used in previous studies and the chosen threshold is often not evaluated.
What is meant by strictly convex?
Strict convexity means that the line segment lies strictly above the graph of f, except at the segment endpoints. (So actually the function in the figure appears to be strictly convex.)
Why convex is important?
Convex functions play an important role in many areas of mathematics. They are especially important in the study of optimization problems where they are distinguished by a number of convenient properties. For instance, a strictly convex function on an open set has no more than one minimum.
Is Huber loss convex?
Huber loss is convex, differentiable, and also robust to outliers.
Why is the loss function convex?
We should always use a convex loss function so that gradient descent can converge to the global minima (local optima free). Neural Networks are very complex non-linear mathematical functions and the loss function most often is non-convex, thus it is usual to stuck in a local minima.
Is the sigmoid convex?
A sigmoid function is convex for values less than a particular point, and it is concave for values greater than that point: in many of the examples here, that point is 0.
What is singular value decomposition (SVD)?
Singular Value Decomposition (SVD) is powerful and ubiquitous tool for matrix factorization but explanations often provide little intuition. My goal is to explain SVD as simply as possible before working towards the formal definition. n imes n n×n unitary matrix. m imes n m×n matrix is a hyperellipse.
What is SVD and why is it important?
It has some interesting algebraic properties and conveys important geometrical and theoretical insights about linear transformations. It also has some important applications in data science. In this article, I will try to explain the mathematical intuition behind SVD and its geometrical meaning.
What is the third definition of a strongly convex function?
A third definition for a strongly convex function, with parameter m, is that, for all x, y in the domain and Notice that this definition approaches the definition for strict convexity as m → 0, and is identical to the definition of a convex function when m = 0.
What is the geometric essence of SVD?
M M of our square (A) can be thought of as simply stretching, compressing, or reflecting that square, provided we rotate the square first (B). This is the geometric essence of SVD. Any linear transformation can be thought of as simply stretching or compressing or flipping a square, provided we are allowed to rotate it first.