Abstract Let f : ℙ 1 → ℙ 1 {f:\mathbb{P}^{1}\to\mathbb{P}^{1}} be a map of degree > 1 {>1} defined over a function field k = K ( X ) {k=K(X)} , where K is a number field and X is a projective curve over K . For each point a ∈ ℙ 1 ( k ) {a\in\mathbb{P}^{1}(k)} satisfying a dynamical stability condition, we prove that the Call–Silverman canonical height for specialization f t {f_{t}} at point a t {a_{t}} , for t ∈ X ( ℚ ¯ ) {t\in X(\overline{\mathbb{Q}})} outside a finite set, induces a Weil height on the curve X ; i.e., we prove the existence of a ℚ {\mathbb{Q}} -divisor D = D f , a {D=D_{f,a}} on X so that the function t ↦ h ^ f t ( a t ) - h D ( t ) {t\mapsto\hat{h}_{f_{t}}(a_{t})-h_{D}(t)} is bounded on X ( ℚ ¯ ) {X(\overline{\mathbb{Q}})} for any choice of Weil height associated to D . We also prove a local version, that the local canonical heights t ↦ λ ^ f t , v ( a t ) {t\mapsto\hat{\lambda}_{f_{t},v}(a_{t})} differ from a Weil function for D by a continuous function on X ( ℂ v ) {X(\mathbb{C}_{v})} , at each place v of the number field K . These results were known for polynomial maps f and all points a ∈ ℙ 1 ( k ) {a\in\mathbb{P}^{1}(k)} without the stability hypothesis,[21, 14],and for maps f that are quotients of endomorphisms of elliptic curves E over k and all points a ∈ ℙ 1 ( k ) {a\in\mathbb{P}^{1}(k)} . [32, 29].Finally, we characterize our stability condition in terms of the geometry of the induced map f ~ : X × ℙ 1 ⇢ X × ℙ 1 {\tilde{f}:X\times\mathbb{P}^{1}\dashrightarrow X\times\mathbb{P}^{1}} over K ; and we prove the existence of relative Néron models for the pair ( f , a ) {(f,a)} , when a is a Fatou point at a place γ of k , where the local canonical height λ ^ f , γ ( a ) {\hat{\lambda}_{f,\gamma}(a)} can be computed as an intersection number.
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Mean Field Analysis of Sparse Reconstruction with Correlated Variables.
Sparse reconstruction algorithms aim to retrieve
high-dimensional sparse signals from a limited number of measurements.
A common example is LASSO or Basis Pursuit where
sparsity is enforced using an `1-penalty together with a cost
function ||y − Hx||_2^2. For random design matrices H, a sharp
phase transition boundary separates the ‘good’ parameter region
where error-free recovery of a sufficiently sparse signal is possible
and a ‘bad’ regime where the recovery fails. However, theoretical
analysis of phase transition boundary of the correlated variables
case lags behind that of uncorrelated variables. Here we use
replica trick from statistical physics to show that when an Ndimensional
signal x is K-sparse and H is M × N dimensional
with the covariance E[H_{ia}H_{jb}] = 1
M C_{ij}D_{ab}, with all D_{aa} = 1,
the perfect recovery occurs at M ∼ ψ_K(D)K log(N/M) in the
very sparse limit, where ψ_K(D) ≥ 1, indicating need for more
observations for the same degree of sparsity.
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- Award ID(s):
- 1344069
- NSF-PAR ID:
- 10019671
- Date Published:
- Journal Name:
- Eusipco
- ISSN:
- 2076-1465
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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