# Spatial Data Reconstruction via ADMM and Spatial Spline Regression +
## 1 Introduction
Our approach penalizes the \"roughness\" of the reconstructed spatial field subject to the constraint that the aggregation of discretized values of the spatial field in each patched subregion equals the aggregate observation made in that subregion. It is distinct from previous spatial smoothing techniques due to the additional constraint in our problem. We propose an Alternating Direction Method of Multipliers (ADMM) [17; 18] algorithm to decouple the problem into the alternated minimizations of a quadratic program (QP) [19] subproblem and a spatial smoothing subproblem, where we use the QP to iteratively enforce the observation constraints, while solving the spatial smoothing subproblem with a recently proposed finite element technique called Spatial Spline Regression (SSR) [16]. In addition, our approach not only leverages the intrinsic smoothness from local continuity to reconstruct a spatial field, but is also able to incorporate additional external information, such as the number of schools, number of bus stops, population, etc., in the underlying geographical region as a regression add-on component to further enhance recovery performance. Last but not least, our algorithm can be applied to a variety of sparse recovery problem where intrinsic smoothness exists. Another important contribution of the paper is that we conduct extensive evaluation to compare our proposed algorithms with a variety of baseline methods. In our evaluation, we are trying to reconstruct the mobile phone activity distributions in Milan, Italy from base station observations. The Telecom Italia Big Data Challenge dataset is a multi-source dataset that contains a variety of informations, including aggregation of telecommunication activities, news, social networks, weather, and electricity data from the city of Milan. With the important information about human activitiescontained in the dataset, especially the cellphone activity records, researchers utilized the data to study different problems, such as modeling human mobility patterns [20; 21; 22], population density estimation [4; 5], models the spread of diseases [23; 24], modeling city structure [3] and city ecology [2], etc. Specifically, our evaluation is based on the Milan Call Detail Records (CDR) dataset, a part of the Telecom Italia Big Data Challenge dataset [25] which contains the phone call and Short Message Service (SMS) activity records of two months in each grid square of 235 m \(\times\) 235 m in the city of Milan, Italy. Given the Milan Call Detail Records (CDR) dataset, we consider a region that consists of 2726 grid squares in an irregularly bounded region in the city of Milan. To stress-test the algorithm performance, we assume we only know the aggregate phone activities observed on 100 or 200 base stations and aim to recover the entire spatial field of phone activities. We also use another geographical attribute dataset available from the Municipality of Milan's Open Data website [1] as the additional external attribute data to improve performance. Extensive evaluation shows that our proposed approach achieves significant improvement, compared to various state-of-the-art baseline methods, including the spatial spline regression (SSR) [16] approach. Our technique can recover the fine-grained cell phone activity distribution of 2726 data points only from 200 data points of base stations, with a mean absolute percentage error of 0.309, representing a 26.3% improvement from the SSR baseline scheme. The remainder of this paper is organized as follows. In Section 2, we formulate the problem of spatial field reconstruction from coarse aggregate observations. In Section 3, we describe existing solutions, including a state-of-the-art Spatial Spline Regression (SSR) technique for spatial smoothing. In Section 4, we propose our Constrained Spatial Smoothing method which respects both the local continuity in the spatial field and the aggregation constraints at the same time.