On Nearly Orthogonal Lattice Bases and ... - Researcher - IBM

More documents

Recommendations

Info

where the orthonormality of X ’s columns is used to obtain (25) from (24). Let y i , i = 1,2,... ,m, and ỹ 1denote column vectors⎡ ⎤ ⎡ ⎤y 1iỹ 11y 2iy i := ⎢ ⎥⎣ . ⎦ and ỹ ỹ 211 := ⎢ ⎥⎣ . ⎦ .y mi ỹ m1Since ỹ 1 = ∑ mk=2 α k y k ,ỹ T 1 y 1 = 0.Then (25) can be rewritten using matrix notation as∣cos ˜θ∣ yT= 1 Ψ 2 ỹ 1 , (26)√y1 TΨ2 y 1√ỹ1 TΨ2 ỹ 1with Ψ 2 := Ψ T Ψ. The angle ˜θ is minimized when the right hand side of (26) is maximized.For arbitrary B with only the singular values known (that is, Ψ is known), the θ-orthogonality of B isgiven by solving the following constrained optimization problem.cos θ = maxy 1 ,ey 1∣ ∣y T1 Ψ 2 ỹ 1∣ ∣√y T 1 Ψ2 y 1√ỹ T 1 Ψ2 ỹ 1, such that ỹ T 1 y 1 = 0. (27)Wielandt’s inequality [14, Thm. 7.4.34] states that if A is a positive definite matrix, with γ min and γ maxdenoting its minimum and maximum eigenvalues (both are positive), then|x T Ay| 2 ≤(γmax − γ minγ max + γ min) 2(x T Ax)(y T Ay)for every pair of orthogonal vectors x and y (equality holds for some pair of orthogonal vectors). In ourproblem, A = Ψ 2 , x = ỹ 1 , y = y 1 , γ max = ψmax 2 and γ min = ψmin 2 . Therefore, using Wielandt’s inequalityand (27), we haveHencewhich proves (20).5.4.2 Proof of Theorem 4cos θ = ψ2 max − ψmin2ψmax 2 + ψmin2 .sin θ = 2ψ maxψ minψmax 2 + , (28)ψ2 min✷The first part of Theorem 4 follows easily. From Section 5.2, we can infer that with m ≤ cn, 0 ≤ c < 1,both ψ min ≥ 1 − √ c and ψ max ≤ 1 + √ c almost surely as n → ∞. Invoking Lemma 2 and substitutingψ min = 1 − √ c and ψ max = 1 + √ c into (20), it follows that as n → ∞, B is θ-orthogonal almost surelywith θ given by (21).14
We now focus on proving the second part of Theorem 4. Let d = √ c, and defineWe first show that for δ ≥ 0,G(d) := 1 − d21 + d 2.G(d + δ) ≥ G(d) − 3√ 3δ. (29)4Using the mean value theorem,(G(d + δ) = G(d) + G ′ d + ˜δ)δ, for some ˜δ ∈ (0,δ), (30)with G ′ denoting the derivative of G with respect to d. Further,G ′ (d) =−4d(1 + d 2 ) 2 ≥ −3√ 3, for d > 0. (31)4One can verify the inequality above by differentiating G ′ (d), and observing that G ′ (d) is minimized when3d 4 + 2d 2 − 1 = 0. The only positive root of this quadratic equation is d 2 = 1/3 or d = 1/ √ 3. Combining(30) and (31), we obtain (29).From the results in Section 5.2, it follows that the probability that both minimum and maximum singularvalues of B satisfy|ψ min | ≥ 1 − (√ c + r + ɛ ) and |ψ max | ≤ 1 + (√ c + r + ɛ ) (32)is greater than 1 − 2e − nr2ρ . When (32) holds, B is at least sin −1 (G( √ c + r + ɛ))-orthogonal. This followsfrom (20). Invoking (29), we can infer that B is θ-orthogonal with θ as in (22).✷6 JPEG Compression History Estimation (CHEst)In this section, we review the JPEG CHEst problem that motivates our study of nearly orthogonal lattices,and describe how we use this paper’s results to solve this problem. We first touch on the topic of digitalcolor image representation and briefly describe the essential components of JPEG image compression.6.1 Digital Color Image RepresentationTraditionally, digital color images are represented by specifying the color of each pixel, the smallestunit of image representation. According to the trichromatic theory [29], three parameters are sufficientto specify any color perceived by humans. 3 For example, a pixel’s color can be conveyed by a vectorw RGB= (w R,w G,w B) ∈ R 3 , where w R, w G, and w Bspecify the intensity of the color’s red (R), green (G),and blue (B) components respectively. Call w RGBthe RGB encoding of a color. RGB encodings are vectorsin the vector space where the R, G, and B colors form the standard unit basis vectors; this coordinate systemis called the RGB color space. A color image with M pixels can be specified using RGB encodings by amatrix P ∈ R 3×M .3 The underlying reason is that the human retina has only three types of receptors that influence color perception.15
Page 1 and 2: On Nearly Orthogonal Lattice Bases
Page 3 and 4: The closest vector problem (CVP) an
Page 5 and 6: (1,2)(1,1.5)90 deg60 deg(0,0)(a)(0,
Page 7 and 8: Invoking Theorem 1 for 2-D lattices
Page 9 and 10: l ∈ {1,2,... ,k}. Then⎡⎤u 11
Page 11 and 12: new properties of random bases and
Page 13: 5.4 Proof of Results on Random Latt
Page 17 and 18: 6.5 Our ApproachOur approach is to
Page 19 and 20: 6.5.4 Experimental ResultsWe tested
Page 21 and 22: [26] W. Pennebaker and J. Mitchell,

On Nearly Orthogonal Lattice Bases and ... - Researcher - IBM

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?