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Incomplete data sets have become almost ubiquitous in a wide variety of application domains. Common examples can be found in climate and image data sets, sensor data sets, and medical data sets. The incompleteness in these data sets may arise from a number of factors: In some cases, it may simply be a reflection of certain measurements not being available at the time, in others, the information may be lost due to partial system failure, or it may simply be a result of users being unwilling to specify attributes due to privacy concerns. When a significant fraction of the entries are missing in all of the attributes, it becomes very difficult to perform any kind of reasonable extrapolation on the original data. For such cases, we introduce the novel idea of conceptual reconstruction in which we create effective conceptual representations on which the data mining algorithms can be directly applied. The attraction behind the idea of conceptual reconstruction is to use the correlation structure of the data in order to express it in terms of concepts rather than the original dimensions. As a result, the reconstruction procedure estimates only those conceptual aspects of the data which can be mined from the incomplete data set, rather than force errors created by extrapolation. We demonstrate the effectiveness of the approach on a variety of real data sets.