ANGULAR RESOLUTION LIMIT FOR DETERMINISTIC CORRELATED SOURCES Xin Zhang∗ , Mohammed Nabil El Korso∗∗ and Marius Pesavento∗ ∗

Communication Systems Group, Technische Universit¨at Darmstadt, Darmstadt, Germany ∗∗ ENS-Cachan / SATIE, Bagneux, France ABSTRACT

This paper is devoted to the analysis of the angular resolution limit (ARL), an important performance measure in the directions-ofarrival estimation theory. The main fruit of our endeavor takes the form of an explicit, analytical expression of this resolution limit, w.r.t. the angular parameters of interest between two closely spaced point sources belonging to the far-field region. As by-products, closed-form expressions of the Cram´er-Rao bound have been derived. Finally, with the aid of numerical tools, we confirm the validity of our derivation and provide a detailed discussion on several enlightening properties of the ARL revealed by our expression, with an emphasis on the impact of the signal correlation. Index Terms— Cram´er-Rao bound, angular resolution limit, Smith’s criterion, directions-of-arrival estimation 1. INTRODUCTION As an important topic within the area of signal processing, far-field source localization in sensor array has found wide-ranging application in, among others, radar, radio astronomy and wireless communications [1]. One common measure to evaluate the performance of this estimation problem is the resolvability of closely spaced signals, in terms of their parameters of interest. In this paper we investigate the minimum angular separation required under which two far-field point sources can still be correctly resolved. To approach this problem, it is necessary to revive the concept of the RL, which will serve as the theoretical cornerstone of this paper. The RL is commonly defined as the minimum distance w.r.t. the parameter of interest (e.g., the directions-of-arrival (DOA) or the electrical angles, etc.), that allows distinguishing between two closely spaced sources [2–4]. Till now there exist three approaches to describe the RL. The first rests on the analysis of the mean null spectrum [5], the second on the detection theory [3, 6], and the third on the estimation theory, capitalizing on the Cram´er-Rao bound (CRB) [2, 7, 8]. A widely accepted criterion based on the third approach, proposed by Smith [2], states that two signals are revolvable if the distance (w.r.t. the parameter of interest) between them is greater than the standard deviation of the distance estimation. In this paper we consider the RL in the Smith’s sense, due to the following reasons: i) it takes the coupling between the parameters into account and thus is preferable to other criteria in the same category, e.g., the one proposed in [7]. ii) A well-known drawback of the mean null spectrum approach is that it is designed for a specific high-resolution algorithm and hence lacks generality. iii) Furthermore, recent evidence shows that the RL based on the detection theory approach is related to that on the Smith’s criterion [4]. This paper investigates the analysis of the RL for two closely spaced correlated deterministic sources. The RL has received recently an increasing interest especially after the publication of

Smith’s paper [2] for which he won the IEEE best paper award. On one hand some prior works on the RL considered some specific criteria as the RL based on hypothesis testing [3, 9–11], or are specific to the MUSIC algorithm [12]. On the other hand prior works based on the Smith criterion are given in non-closed form expressions as in [6, 13] or are given for some specific assumptions (e.g., one known DOA [2, 14], uncorrelated sources [12], ULA case [2, 8, 14], non time varying sources [8] etc.) In our work, we propose to derive an analytical expression for the angular resolution limit (ARL)1 , denoted by δ, between two closely spaced, time-varying (both in amplitude and phase) far-field point sources impinging on non-uniform linear array, which, to the best of our knowledge, is till now absent in the current literature. Furthermore, our expression, by virtue of its concise form highlighting the respective effects of various factors on δ, reveals a number of enlightening properties pertinent to the ARL’s behavior (e.g., concerning the source correlation), while being also computationally efficient by saving the trouble of solving a non-linear equation numerically. The following notation will be used throughout this paper: (·)H , T (·) denote the conjugate transpose and the transpose of a matrix, respectively. tr{·} and vec{·} denote the trace and the vectorization of a matrix, respectively. <{·} and ={·} denote the real and imaginary part respectively. k·k denotes the norm of a vector, ⊗ denotes the Kronecker product, whereas E{·} denotes expectation. 2. MODEL SETUP Consider a linear, possibly non-uniform, array comprising M sensors that receives two narrowband time-varying far-field sources s1 (t) and s2 (t), the directions-of-arrival of which are θ1 and θ2 , respectively. Then the received signal at the m-th sensor can be expressed as [1]: xm (t) =

2 X

si (t)ejkdm sin(θi ) + nm (t),

i=1

t = 1, . . . , N

(1)

and m = 1, . . . , M. where the sources are modeled by2 si (t) = ai (t)ej(2πf0 +πi (t)) , i = 1, 2 in which ai (t) denotes the time-varying non-zero real amplitude, f0 denotes the carrier frequency, and πi (t) denotes the timevarying phase; dm denotes the spacing between the first sensor (which is chosen as the so-called reference sensor, i.e., d1 = 0) and is the wave number (with λ denoting the wave the m-th, k = 2π λ length), nm (t) denotes the additive noise at the m-th sensor, and N is the number of snapshots. 1 The so-called ARL characterizes the RL when we consider the angular parameters as the unknown parameters of interest. 2 Note that this is a commonly used signal model in communication systems (cf. [15, 16]).

Fore mathematical convenience, define νi = k sin(θi ), i = 1, 2 where ∆ = ν1 − ν2 denotes the spacing between ν1 and ν2 . (Asas our parameters of interest. Changing (1) into the vector form, one sumption is made here that ν1 > ν2 .) ¯ we obtain the following CRBs: obtains x(t) = As(t) + n(t), where x(t) = [x1 (t), . . . , xM (t)]T , By inverting the 2 × 2 matrix I h i s(t) = [s1 (t), s2 (t)]T , n(t) = [n1 (t), . . . , nM (t)]T , and A = 2N α ¯ −1 = SNR2 , (7) [a(ν1 ), a(ν2 )]. The steering vector are defined as a(νi ) = [ejνi d1 , . . . , ejνi dM ]T , i = CRB(ν1 ) , I Ψ 1,1 1, 2. Furthermore, define the correlation factor ρ between the two h i 2N α signals as [17] ¯ −1 CRB(ν2 ) , I = SNR1 , (8) H Ψ 2,2 s 1 s2 ρ= (2) and ks1 k ks2 k h i 2 ¯ −1 (9) CRB(ν = − 2 <{η}. T 1 , ν2 ) , I where si = [si (1), . . . , si (N )] , i = 1, 2 are signal vectors. σ Ψ 1,2 The following assumptions are made in the remaining of this 2 2 4 2 where Ψ = 4α N SNR1 ·SNR2 −(4/σ )·< {η} is the determinant chapter: of I. A1 The sensor noise follows a complex circular white Gaussian distributed, both spatially and temporally, with zero-mean and 3.2. Equation Setup and Simplification unknown noise variance σ 2 . According to the Smith’s criterion, the ARL, δ, is given as the anA2 The source signals are assumed to be deterministic; and the sepgular spacing, ∆, which is equal to the standard deviation of the aration of the sources is small. estimation of ∆. p The latter, under mild conditions [19], can be ap T proximated as CRB(∆), suggesting that δ can be obtained as the A3 The unknown parameter vector is ξ = ν1 , ν2 , σ 2 . Thus, for (positive) solution of the following equation: given ξ, the joint probability density function of the observation χ = [xT (1), . . . , xT (N )]T can be written as p(χ |  δ 2 = CRB(δ). (10) ξ) = πM N1 |R| exp −(χ − µ)H R−1 (χ − µ) , where R = T  where CRB(δ) = CRB(ν1 ) + CRB(ν2 ) − 2CRB(ν1 , ν2 ) [4]. σ 2 I M N and µ = (As(1))T , . . . , (As(N ))T . Substituting (7)-(9) into (10), the latter is transformed into: 3. DERIVATION OF δ The derivation of the ARL δ can be divided into three steps. The first step involves the derivation of the CRBs w.r.t. the relevant parameters. The second builds on this result and simplifies the implicit function based on the Smith’s criterion, the root of which yields δ. The last step is to solve the function corresponding to different values of ρ, leading to the obtention of the ARL.

δ 2 = CRB(ν1 ) + CRB(ν2 ) − 2CRB(ν1 , ν2 ) (11) 2 2 = (N · SNR2 α + N · SNR1 α + 2 <{η}). Ψ σ Substituting δ for ∆ in (6) identity we observe that (11) is a highly non-linear equation in δ. Hence, in order to obtain the solution of (11) w.r.t. δ, and taking into account that δ is small, we resort to the first-order Taylor’s expansion of η around δ = 03 to obtain: η ≈ sH 1 s2

3.1. CRB Derivation

d2m (1 − jdm δ)

m=1

The CRB of the unknown parameters (ν1 and ν2 ) can be obtained by the analytical inverse of the Fisher information matrix (FIM) for ξ (denoted by I). Under Gaussian noise, the elements of I can be calculated using the following formula [18]: ( ) ( ) ∂R −1 ∂R ∂µH −1 ∂µ [I]i,j = tr R−1 R + 2< R . ∂ [ξ]i ∂ [ξ]j ∂ [ξ]i ∂ [ξ]j (3) where [ξ]i denotes the i-th element of ξ. Thus for our model, I takes the following block-diagonal form:   ¯ I 0 I= , (4) 0T Mσ4N

M X

=

sH 1 s2

=

sH 1 s2 (α

d2m

− jδ

m=1

where β =

PM

3 m=1 dm .

M X

! d3m

(12)

m=1

− jδβ),

Combining (12) with (2), it follows:

<{η} ≈ ks1 k ks2 k <{ρ(α − jδβ)} = N ε1 ε2 (¯ ρα + ρ˜βδ),

(13)

where ρ¯,˜ ρ are defined as the real and imaginary part of ρ, respectively, i.e., ρ¯ = <{ρ} and ρ˜ = I{ρ}. Now we merge (13) into (11) and, after some mathematical manipulations, obtain the following quartic function of δ: A+B + C = 0, (14) 2 where A, B, C and SNR1 α, B = N · √ D are defined as A = N · √ SNR2 α, C = N SNR1 · SNR2 ρ¯α, and D = N SNR1 · SNR2 ρ˜β. Thus our task of finding the expression of δ has been brought down to finding the root of (14). D2 δ 4 + 2CDδ 3 + (C 2 − AB)δ 2 + Dδ +

where ¯= I



2N αSNR1 2 <{η} σ2

2 <{η} σ2

2N αSNR2

 ,

(5)

P 2 2 2 in which α = M m=1 dm , SNRi = εi /σ , i = 1, 2, where εi = qP N 2 t=1 ai (t)/N , i = 1, 2; and η = sH 1 s2

M X

M X m=1

d2m e−jdm (ν1 −ν2 ) = sH 1 s2

M X m=1

d2m e−jdm ∆ , (6)

3 In asymptotic cases δ becomes very small, and our approximation made here is tight, as will be proved by our simulation (cf. Fig. 1). This can be explained by the fact that the maximum likelihood estimator, and generally all high resolution estimators, have an asymptotically infinite capability of resolution leading to δ → 0.

3.3. Expression of δ The solution of (14), depending on different values of ρ, falls into the following three cases: Case 1. Non-zero imaginary part of the correlation coefficient ρ (˜ ρ 6= 0): in this case (14) remains a quartic function in δ. We know from the parameter transformation property of the CRB (cf. [20], p.37) that CRB(δ) = CRB(−δ), Thus, if δ is a root of (10) (hence of (14)), then −δ will also be a root thereof, which allows us to remove all the terms of odd degrees in (14), leading to a quadratic equation of δ 2 : D2 δ 4 + (C 2 − AB)δ 2 +

A+B + C = 0, 2

(15)

the root of which is4 : AB − C 2 −

2

q

δ =

(C 2 − AB)2 − 4D2 ( A+B + C) 2 2D2

γ = κ

s 1−

ακφ 1− 2 γ

, (16)

(17)

The existence of δ 2 in (16) is assured since under realistic conditions (ακφ/γ 2 )  1. Thus the ARL is given by: v s ! u uγ ακφ t δ= 1− 1− 2 . (18) κ γ Case 2. Not fully correlated signals with zero imaginary part of the correlation coefficient ρ (˜ ρ = 0 and ρ¯ 6= ±1): in this case D = 0, (C 2 − AB) 6= 0, and (14) degenerates into (C 2 − + C = 0. Taking its positive root we have: AB)δ 2 + A+B 2 s s A+B +C φα 2 δ= = . (19) AB − C 2 2γ The existence of δ is guaranteed from the fact that in this case both φ and γ are greater than zero. It is worth noticing that an important special case of Case 2, in which both ρ˜ and ρ¯ equal zero, namely, the two signals are uncorrelated, reduces (19) q to δ =

1 (1/SNR1 2N α

Note that, for the Uniform Linear Array (ULA) configuration, α = 2 2 M (M −1)(2M −1) d and β = M (M4 −1) d, where d denotes the inter6 sensor spacing. 4. SIMULATIONS AND NUMERICAL ANALYSIS The context of our simulation is a ULA of comprising M = 6 sensors with half-wave length inter-element spacing. The snapshot number is given by N = 100. Our results are as follows:

!

where γ = (1 − ρ¯2 )α2 , κ = 2˜ ρ2 β 2 and   1 1 1 2¯ ρ φ= + +√ . N SNR1 SNR2 SNR1 · SNR2

Now, combining the results in all three cases with which we have dealt in turn, our final expression of the ARL can be written as:  v s ! u   uγ  ακφ  t  1 − 1 − , for ρ˜ 6= 0   κ γ2  s δ=  φα   , for ρ˜ = 0 and ρ¯ 6= ±1   2γ    (no closed-form expression available), for ρ˜ = 0 and ρ¯ = ±1 (20)

+ 1/SNR2 ).

Case 3. Fully correlated signals with zero imaginary part of the correlation coefficient ρ (˜ ρ = 0 and ρ¯ = ±1): in this case (14) degenerates to A+B + C = 0 and have no root any 2 more.5 4 The other root of (15), which is very large, is in contradiction with the observation made in Footnote 3, thus is regarded as trivial and rejected. 5 One can expect that for this case, in which ρ = ±1, i.e., the two signals are linearly dependent, the approximation made using a first order Taylor’s expansion would not be sufficiently tight w.r.t. the true model. Nevertheless, after lengthy calculus, we noticed that using a higher order Taylor’s expansion leads to the same observation, i.e., that no closed-form expression can be deduced for ρ = 1. As will be shown in the simulation part, this suggests that the observation model would no be identifiable any more. The ρ = −1 case, on the other hand, involves solving a sextic equation, the detailed analysis of which, unfortunately, is due to the space limitation beyond the scope of this paper.

• In Fig. 1 we validate our approximate analytical expression of δ in (20) for two cases (˜ ρ 6= 0; ρ˜ = 0&¯ ρ 6= 1) by comparing it with the true δ (obtained by solving (10) numerically) and showing that they are identical. • As is revealed by (20), the concrete waveforms of the signals has no effect on δ, which only depends on the two signal’s respective strengths (ε1 , ε2 ) and the correlation ρ between them. Furthermore, note that either ρ¯ or ρ˜ plays its role separately. Fig. 2 shows that, with a fixed ρ¯, δ gets slightly higher as the value of |˜ ρ| increases. But this impact is so limited compared with the that of ρ¯, that the former is totally negligible, (cf. Fig. 3 and Fig. 4, both of which show that δ increases notably as ρ¯ raises, while remains nearly unreactive to the change of ρ˜.) This fact can be explained by noticing that, since (ακφ/γ 2 )  1, one can apply the first order Taylor’s expansion to (18) around (ακφ/γ 2 ) = 0 and obtains: s    s γ ακφ φα δ≈ 1− 1− = , (21) κ 2γ 2 2γ which is the same expression as (19), independent of ρ˜. • Fig. 5 casts light ρ = ±1 (linearly dependent signals) cases, which demonstrates that δ → ∞ as ρ → 1, however we enhance the number of snapshot or signal strengths, suggesting that two perfect positive linear signals can by no means be resolved (suggesting that the observation model could in this case be non-identifiable); meanwhile, δ approaches its lowest value when ρ → −1 (we also corroborated this fact by adding in the true values of δ for ρ = −1 under various contexts, acquired by numerically solving (11)), which means that two signals enjoy a minimum ARL between them if they have a perfect negative linear relationship. • The dependence of δ on signal strengths is reflected in the expression of φ (cf. (17)), where we see that if the strength of one signal is much greater than the other, e.g., ε1  ε2 , then φ ≈ 1/(N · SNR2 ), and δ becomes restricted by the weaker signal. Thus enhancing the strength of only one signal cannot infinitely diminish δ, as is shown by Fig. 6, in which we increase ε1 from 1 to 1000 while keeping ε2 = 1, and find that δ converges to a certain value (determined by ε2 ). Fig. 6 also

investigates the impact of the sensor array geometry on δ (cf. Table 1) and reveals that a loss of sensors in the array configuration has a considerable impact on δ only when it causes a diminution of the aperture size of the array, as in the case of Type 1. If, however, the array aperture remains unchanged, as in the case of Type 2, this impact is considerably mitigated.

0.025 Re{ρ}=−0.7 Re{ρ}=−0.3 Re{ρ}=0 Re{ρ}=0.3 Re{ρ}=0.7

0.02

δ 0.015

Array Type

Geometric Configuration 0.01 0

◦ • • ◦ • • ◦ ◦ • ◦ ◦ ◦ ◦ • • • • • • • • • • •

Type 1 Type 2 Type 3

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

|Im{ρ}|

Fig. 3. δ vs. |˜ ρ| for ε1 = ε2 = 1, σ 2 = 1 and various ρ¯.

Table 1. Different array geometric configurations. • and ◦ represent the position of sensor and missing sensors, respectively. The interelement spacing is half-wave length.

1

10

|Im{δ}|=0 |Im{δ}|=0.2 |Im{δ}|=0.5 |Im{δ}|=0.8

0

10

δ

−1

10

−1

10

−2

10 −2

10

−3

10

δ

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Re{ρ} Analytical (approximate) δ, ρ=0.5+0,5j Numerical (true) δ, ρ=0.5+0,5j Analytical (approximate) δ, ρ=−0.5 Numerical (true) δ, ρ=−0.5

−3

10

Fig. 4. δ vs. ρ¯ for ε1 = ε2 = 1, σ 2 = 1 and various ρ˜.

−4

10

−3

−2

10

−1

10

0

10 2

1

10

N=100, ε1=ε2=1

10

σ

N=100, ε1=ε2=1,ρ=−1

2

10

N=500, ε1=ε2=10

Fig. 1. Numerical and analytical δ vs. σ 2 for ε1 = ε2 = 1, with ρ = 0.5 + 0.5j and ρ = −0.5 respectively.

N=500, ε1=ε2=10, ρ=−1

0

δ

10

N=1000, ε1=ε2=100 N=1000, ε1=ε2=100, ρ=−1

−2

10

−4

5. SUMMARY

10

−1

In this paper we studied the angular resolution between two point sources and provide a closed-form expression of δ, the validity of which was confirmed by simulation. We also noticed that δ is not dependent on the special waveforms of the signals, but only on their strengths and the correlation factor between them, and that the imaginary part of ρ only has a negligible impact on δ, while the impact of the real part of ρ is decisive. We also examined δ for ρ = ±1 cases. Finally, we showed that δ is constrained by the weaker sig-

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Re{ρ}

Fig. 5. δ vs. ρ¯ for ρ˜ = 0, with various N , ε1 and ε2 . 0.025 Type 1 Type 2 Type 3

0.02

δ

0.015

0.0191 0.01 0.0191

δ

0.005 0 10

0.0191

1

10

2

ε2

10

3

10

0.0191

Fig. 6. δ vs. ε2 for ρ = 0.5 + 0.5j, ε1 = 1, σ 2 = 1 and various array configurations.

0.0191

0.019 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

|Im{ρ}| 2

Fig. 2. δ vs. |˜ ρ| for ρ¯ = 0.5, ε1 = ε2 = 1 and σ = 1.

0.9

nal, thereby cannot be infinitely decreased. Finally the impact of different array geometries on δ is discussed.

References [1] H. Krim and M. Viberg, “Two decades of array signal processing research: the parametric approach,” IEEE Signal Processing Mag., vol. 13, no. 4, pp. 67–94, 1996. [2] S. T. Smith, “Statistical resolution limits and the complexified Cram´er Rao bound,” IEEE Trans. Signal Processing, vol. 53, pp. 1597–1609, May 2005. [3] M. Shahram and P. Milanfar, “On the resolvability of sinusoids with nearby frequencies in the presence of noise,” IEEE Trans. Signal Processing, vol. 53, no. 7, pp. 2579–2588, July 2005. [4] M. N. El Korso, R. Boyer, A. Renaux, and S. Marcos, “Statistical resolution limit for multiple parameters of interest and for multiple signals,” in Proc. ICASSP, Dallas, Texas, USA, Mar. 2010, pp. 3602–3605. [5] H. Cox, “Resolving power and sensitivity to mismatch of optimum array processors,” J. Acoust. Soc., vol. 54, no. 3, pp. 771–785, 1973. [6] Z. Liu and A. Nehorai, “Statistical angular resolution limit for point sources,” IEEE Trans. Signal Processing, vol. 55, no. 11, pp. 5521–5527, Nov. 2007.

[11] D. T. Vu, M. N. El Korso, R. Boyer, A. Renaux, and S. Marcos, “Angular resolution limit for vector-sensor arrays: detection and information theory approaches,” in Proc. IEEE Workshop on Statistical Signal Processing, SSP-2011, Nice, France, June 2011, pp. 9–12. [12] H. Abeida and J.-P. Delmas, “Statistical performance of MUSIC-like algorithms in resolving noncircular sources,” IEEE Trans. Signal Processing, vol. 56, no. 6, pp. 4317–4329, Sept. 2008. [13] Y.I. Abramovich, B.A. Johnson, and N.K. Spencer, “Statistical nonidentifiability of close emitters: Maximum-likelihood estimation breakdown,” in EUSIPCO, Glasgow, Scotland, Aug. 2009. [14] R. Boyer, “Performance bounds and angular resolution limit for the moving co-located MIMO radar,” IEEE Trans. Signal Processing, vol. 59, no. 4, pp. 1539–1552, Apr. 2011. [15] L.C. Godara, “Applications of antenna arrays to mobile communications: II. Beam-forming and direction of arrival considerations,” IEEE Trans. Antennas Propagat., vol. 85, no. 8, pp. 1195–1245, Aug. 1997.

[7] H. B. Lee, “The Cram´er-Rao bound on frequency estimates of signals closely spaced in frequency,” IEEE Trans. Signal Processing, vol. 40, no. 6, pp. 1507–1517, 1992.

[16] J. Li, P. Stoica, and D. Zheng, “Efficient direction and polarization estimation with a cold array,” IEEE Trans. Antennas Propagat., vol. 44, no. 4, pp. 539–547, Apr. 1996.

[8] M. N. El Korso, R. Boyer, A. Renaux, and S. Marcos, “Statistical resolution limit of the uniform linear cocentered orthogonal loop and dipole array,” IEEE Trans. Signal Processing, vol. 59, no. 1, pp. 425–431, Jan. 2011.

[17] F. R¨omer and M. Haardt, “Deterministic Cram´er-Rao bounds for strict sense non-circular sources,” in Proc. ITG/IEEE Workshop on Smart Antennas (WSA’07), Vienna, Austria, Feb. 2007.

[9] A. Amar and A.J. Weiss, “Fundamental limitations on the resolution of deterministic signals,” IEEE Trans. Signal Processing, vol. 56, no. 11, pp. 5309–5318, Nov. 2008. [10] M. N. El Korso, R. Boyer, A. Renaux, and S. Marcos, “Statistical resolution limit for source localization with clutter interference in a MIMO radar context,” IEEE Trans. Signal Processing, vol. 60, no. 5, pp. 987–992, Feb. 2012.

[18] P. Stoica and R.L. Moses, Spectral Analysis of Signals, Prentice Hall, NJ, 2005. [19] E. L. Lehmann, Theory of Point Estimation, Wiley, New York, 1983. [20] S. M. Kay, Fundamentals of Statistical Signal Processing : Estimation Theory, vol. 1, Prentice Hall, NJ, 1993.

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Semi-deterministic urban canyon models of received power for ...
Urban Canyon Model. CWI Model. 1795. Page 2 of 2. Semi-deterministic urban canyon models of received power for microcells.pdf. Semi-deterministic urban ...

Min Max Generalization for Deterministic Batch Mode ... - Orbi (ULg)
Nov 29, 2013 - One can define the sets of Lipschitz continuous functions ... R. Fonteneau, S.A. Murphy, L. Wehenkel and D. Ernst. Agents and Artificial.

Deterministic Performance Bounds on the Mean Square Error for Near ...
the most popular tool [11]. However ... Date of publication November 27, 2012; ... of this manuscript and approving it for publication was Dr. Benoit Champagne.

Deterministic Algorithms for Matching and Packing ...
Given a universe U, and an r-uniform family F ⊆ 2U , the (r, k)-SP problem asks if ..... sets-based algorithms can speed-up when used in conjunction with more ..... this end, for a matching M, we let M12 denote the subset of (U1 ∪ U2) obtained.

Min Max Generalization for Deterministic Batch Mode ... - Orbi (ULg)
Electrical Engineering and Computer Science Department. University of Liège, Belgium. November, 29th, 2013. Maastricht, The Nederlands ...

Inference Protocols for Coreference Resolution - GitHub
R. 23 other. 0.05 per. 0.85 loc. 0.10 other. 0.05 per. 0.50 loc. 0.45 other. 0.10 per .... search 3 --search_alpha 1e-4 --search_rollout oracle --passes 2 --holdout_off.