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... one can resort to the regularization inductive principle, where a linear combination of a fitting cost Remp, called empirical risk, and a smoothness-enforcing penalty J , is minimized as (see, e.g., =-=[17]-=-) minimize l∈S Remp(l; {(x,φx, yx)}x∈X ) + λJ(l). (9) Here, S is the space of functions, assumed to be a RKHS of vector-valued functions from Rd to RM [10], and λ > 0 is a constant adjusted to attain ...

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...e ‖l‖S is the norm induced by the inner product of S. The empirical risk Remp is chosen to linearly penalize deviations from (8), using the so-called -insensitive loss u(y) := max(0, |y| − ) as in =-=[18]-=- Remp(l; {(x,φx, yx)}x∈X ) := 1 N ∑ x∈X u(yx − φTx l(x)). (10) This is a convex surrogate for the number of measurement errors, in the same way as the `1-norm is substituted for the `0-norm in the sp...

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...e spatial power distribution of each channel. Spatial interpolation is accomplished using an online nonparametric regression technique that hinges on the framework of RKHSs of vector-valued functions =-=[10]-=-, where the coefficients of the frequency basis functions are estimated as spatial vector fields. This work was partially funded by the Spanish Government and the European Regional Development Fund (E...

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...sing Kriging [2], [3], orthogonal matching pursuit [4], sparse Bayesian learning [5], and dictionary learning [6]. To account for the frequency dimension, a basis expansion model (BEM) was adopted in =-=[7]-=-, [8]. However, these techniques require exchanging raw measurements, which imposes stringent requirements on the control channel. This issue was mitigated in [9], but the spatial dimension was not ac...

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...on, network planning, and in particular, in dynamic spectrum access (DSA) of cognitive radios, which aspire to opportunistically exploit underutilized spectral resources in time, space, and frequency =-=[1]-=-. Spatial interpolation of RF power measurements has been tackled using Kriging [2], [3], orthogonal matching pursuit [4], sparse Bayesian learning [5], and dictionary learning [6]. To account for the...

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...adios, which aspire to opportunistically exploit underutilized spectral resources in time, space, and frequency [1]. Spatial interpolation of RF power measurements has been tackled using Kriging [2], =-=[3]-=-, orthogonal matching pursuit [4], sparse Bayesian learning [5], and dictionary learning [6]. To account for the frequency dimension, a basis expansion model (BEM) was adopted in [7], [8]. However, th...

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...Kriging [2], [3], orthogonal matching pursuit [4], sparse Bayesian learning [5], and dictionary learning [6]. To account for the frequency dimension, a basis expansion model (BEM) was adopted in [7], =-=[8]-=-. However, these techniques require exchanging raw measurements, which imposes stringent requirements on the control channel. This issue was mitigated in [9], but the spatial dimension was not account...

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...imate the PSD map (2). In order to allow for estimation in wide frequency bands at low hardware cost and power consumption, acquisition via analog-to-information converters (AICs) is considered [13]– =-=[15]-=-. For each input block rx[b] := [rx[bL], . . . , rx[bL+(L− 1)]]T of L samples rx[k] := rx(kTs), where 1/Ts is the Nyquist rate, an AIC produces a linearly compressed block r̃x[b] of L̃ (< L) samples. ...

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...ive radios, which aspire to opportunistically exploit underutilized spectral resources in time, space, and frequency [1]. Spatial interpolation of RF power measurements has been tackled using Kriging =-=[2]-=-, [3], orthogonal matching pursuit [4], sparse Bayesian learning [5], and dictionary learning [6]. To account for the frequency dimension, a basis expansion model (BEM) was adopted in [7], [8]. Howeve...

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...to estimate the PSD map (2). In order to allow for estimation in wide frequency bands at low hardware cost and power consumption, acquisition via analog-to-information converters (AICs) is considered =-=[13]-=-– [15]. For each input block rx[b] := [rx[bL], . . . , rx[bL+(L− 1)]]T of L samples rx[k] := rx(kTs), where 1/Ts is the Nyquist rate, an AIC produces a linearly compressed block r̃x[b] of L̃ (< L) sam...

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... expansion model (BEM) was adopted in [7], [8]. However, these techniques require exchanging raw measurements, which imposes stringent requirements on the control channel. This issue was mitigated in =-=[9]-=-, but the spatial dimension was not accounted for. The aim of the present work is to address these limitations through an estimator-interpolator approach that can afford lowcost, low-power sensing arc...

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...ems obey transmission standards (e.g. DVB or ATSC in TV bands) and spectrum mask regulations, which fix the transmission parameters, such as bandwidth, carrier frequencies, and roll-off factors [11], =-=[12]-=-. If the functions φm(f), m = 1, . . . ,M − 1, are not known, our approach can still be used by adopting a general BEM [7], [8]. The received waveform at position x ∈ R, due to the M−1 uncorrelated tr...

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...y systems obey transmission standards (e.g. DVB or ATSC in TV bands) and spectrum mask regulations, which fix the transmission parameters, such as bandwidth, carrier frequencies, and roll-off factors =-=[11]-=-, [12]. If the functions φm(f), m = 1, . . . ,M − 1, are not known, our approach can still be used by adopting a general BEM [7], [8]. The received waveform at position x ∈ R, due to the M−1 uncorrela...

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...pace, and frequency [1]. Spatial interpolation of RF power measurements has been tackled using Kriging [2], [3], orthogonal matching pursuit [4], sparse Bayesian learning [5], and dictionary learning =-=[6]-=-. To account for the frequency dimension, a basis expansion model (BEM) was adopted in [7], [8]. However, these techniques require exchanging raw measurements, which imposes stringent requirements on ...

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...tically exploit underutilized spectral resources in time, space, and frequency [1]. Spatial interpolation of RF power measurements has been tackled using Kriging [2], [3], orthogonal matching pursuit =-=[4]-=-, sparse Bayesian learning [5], and dictionary learning [6]. To account for the frequency dimension, a basis expansion model (BEM) was adopted in [7], [8]. However, these techniques require exchanging...

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...his operation can be represented as r̃x[b] = G̃rx[b] (3) where G̃ ∈ CL̃×L is the compression matrix. In order to guarantee the identifiability of the lm(x)’s, this matrix must satisfy the criteria in =-=[16]-=-. The whole observation frame rx := [rTx [0], . . . , r T x [B − 1]]T therefore produces r̃x = Grx, where r̃x := [r̃Tx [0], . . . , r̃ T x [B − 1]]T and G := IB ⊗ G̃ (⊗ denotes the Kronecker product)....

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...d certain requirements are met [10]. A simple construction is K(z,x) = diag { k(1)(z,x), . . . , k(M)(z,x) } , where k(m)(z,x) are valid scalar kernels. We developed a batch algorithm to solve (9) in =-=[19]-=-. However, the cost that batch approaches incur can grow prohibitively as the number of measurements increases. This motivates well the online algorithm here, which can incrementally update the estima...

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...spectral resources in time, space, and frequency [1]. Spatial interpolation of RF power measurements has been tackled using Kriging [2], [3], orthogonal matching pursuit [4], sparse Bayesian learning =-=[5]-=-, and dictionary learning [6]. To account for the frequency dimension, a basis expansion model (BEM) was adopted in [7], [8]. However, these techniques require exchanging raw measurements, which impos...

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...s can track slow temporal variations of the fields of interest. An elegant approach to derive online algorithms for kernel machines is based on stochastic gradient descent in the function space [20], =-=[21]-=-. First, one can define the instantaneous regularized error as Rinst(l,φ,x, y) := u(y − φT l(x)) + λ‖l‖2S . (11) Note that the objective of (9) is just the sample average of Rinst. Suppose that per s...