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3261Area of an Ellipse from the Unit DiskThe map x=2u,\ y=3v sends the unit disk u 2+v 2\le 1 to an ellipse in the (x,y)-plane. What is the area of that ellipse?数学中等derivation未尝试面试订阅3262Area of a Parallelogram from a Unit SquareThe map x=u+v,\ y=u-v sends the unit square 0\le u,v\le 1 to a parallelogram. What is its area?数学中等derivation未尝试面试订阅3263Integral After a Simple ScalingUse the change of variables x=2u,\ y=3v to compute \iint R 1\,dx\,dy, where R is the image of the rectangle 0\le u\le 1,\ 0\le v\le 2.数学中等derivation未尝试面试订阅3264Area Scaling Under a Triangular MapThe map x=u,\ y=u+2v sends the unit square to a parallelogram. Compute the image area.数学中等derivation未尝试面试订阅3265Area of a Polar WedgeUse the polar Jacobian to compute the area of the region 0\le r\le 2,\ 0\le \le /3.数学中等derivation未尝试面试订阅3266Best Rank-One Reconstruction Error From Singular ValuesA centered data matrix has singular values \sigma 1\ge\sigma 2\ge\cdots\ge\sigma r. What is the Frobenius-norm reconstruction error of the best rank-one approximation?数学中等derivation未尝试面试订阅3268Recovering Correlation Strength from a Known First PCA covariance matrix has the form \Sigma=egin pmatrix 5&c\c&5\end pmatrix . You are told that the first principal component points along (1,1) and has variance 8. What is c?数学中等derivation未尝试面试订阅3269One-PC Reconstruction of a Two-Asset MoveLet \Sigma=egin pmatrix 9&-3\-3&9\end pmatrix , whose first principal direction is along (1,-1). For the observed move x=(2,-1), what are the rank-1 reconstruction using only the first principal component and the residual?数学中等derivation未尝试面试订阅3270How Standardization Changes the PCA SpectrumA raw covariance matrix is \Sigma=egin pmatrix 9&6\6&9\end pmatrix . After standardizing each coordinate to unit variance, what are the eigenvalues of the correlation matrix, and what fraction of standardized variance does the first component explain?数学中等derivation未尝试面试订阅3272How Large Can the Third Singular Value Be?A data matrix has singular values 7, 4, and s. You plan to keep rank 2 and want to retain at least 95\% of total squared Frobenius energy. What is the largest allowable value of s?数学困难derivation未尝试面试订阅3273Which Portfolio Is Better Aligned with the Low-Variance PC?A covariance matrix has eigenvalues 25 and 1, with first eigenvector proportional to (2,1) and second eigenvector proportional to (1,-2). Compare the variances of portfolios p 1=(1,-2) and p 2=(2,1).数学中等derivation未尝试面试订阅3277Covariance of PCA ScoresIf data are centered and projected onto orthonormal eigenvectors of the covariance matrix, what is the covariance matrix of the resulting PCA score vector?数学中等derivation未尝试面试订阅3278How Large Must the Leading Eigenvalue Be to Reach 75%?A covariance matrix has eigenvalues \lambda 1, 3, and 2. What is the smallest value of \lambda 1 such that the first principal component captures at least 75\% of total variance?数学中等derivation未尝试面试订阅3279Recovering the Tail Singular Value from an Energy StatementA matrix has Frobenius norm 10, and its first two singular directions explain exactly 24/25 of total squared Frobenius energy. What is the smallest singular value?数学中等derivation未尝试面试订阅3280What Does Whitening Mean When the Covariance Is Singular?Suppose \Sigma=egin pmatrix 2&2\2&2\end pmatrix . Explain what can and cannot be whitened, and identify the subspace where a pseudo-whitening transform still makes sense.数学中等essay未尝试面试订阅3281How Much of a Single-Asset Position Comes from the Market PC?Under covariance \Sigma=egin pmatrix 5&4\4&5\end pmatrix , take the portfolio p=(1,0). What fraction of its variance is attributable to the first principal component?数学中等derivation未尝试面试订阅3282Residualizing a Move Against the First PCUsing the same first principal direction u=(1,1)/\sqrt2, remove the first-PC component from the observed move r=(1,3). What residual vector remains?数学中等derivation未尝试面试订阅3283Which Singular Direction Blows Up Noise the Most?A design matrix has singular values 12, 5, and 0.5. Under the pseudoinverse, by what factor is noise amplified in the smallest-singular-value direction relative to the largest-singular-value direction?数学中等derivation未尝试面试订阅3284Why Centering Matters Before PCAWhy can skipping mean-centering make the first principal component mostly track the mean level instead of genuine variation?数学中等essay未尝试面试订阅3285Why Whitening Can Hurt Noisy Small-Eigenvalue DirectionsWhy can whitening make downstream models more fragile when some covariance eigenvalues are tiny?数学中等essay未尝试面试订阅